Cell biology improvement drive

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'''Part of  [[:Category:Social Sciences|Social Sciences]]'''

==Introduction and Descripton==

'''Welcome to Wikiversity's Department of Women's Studies'''

Hello! The Wikiversity Department of Women's Studies has just been founded. If you are knowledgable about any of the topics listed below, or about other Women's Studies topics, your help would be extremely appreciated if you make a contribution.

Please do not delete learning projects. Instead, try to add a project if you would like it to be offered or if you would like to offer it.

==Learning Resources==
{{Template:Courses}}

===Core Lessons===

* [[Introduction to Wiki]] '''AND'''
* [[/Women in Perspective/]] '''OR'''
* [[/International Perspectives on Women/]] '''AND'''
* [[/Feminist Theory/]]

===Tracks===

'''Please Choose Any Section Below:'''

==== Multiculturalism and Diversity ====

* [[/International Perspectives on Women/]]
* [[/Women of Color/]]
* [[/Trans Perspectives in the Early 21st Century/]]
* [[/Gender in Cross-Cultural Perspective/]]
* [[/Latinas in the Americas/]]
* [[/World Literature: African Women Writers/]]
* [[/African-American Women Writers/]]
* [[Gays and Lesbians in U.S. History]]
* [[Queer Studies]]
* [[/Life Histories of African Women/]]
* [[/Race, Class and Gender/]]
* [[/Women in World Religions/]]
* [[/Psychology of Lesbian Culture/]]
* [[/Sociology of Race, Class and Gender/]]

=== Suggested tracks ===

==== Nature; Science ====

'''Suggested courses for this track:'''

*Reproductive Technologies and the Future of Motherhood
*Biology of Women
*Economic Issues of Gender
*Women’s Health
*Perspectives on Rape and Sexual Assault
*Women, Gender and Science
*Women, Society and Radiation
*The American Woman in Sport
*Special Topics: Women and Law
*Sex Differences: Psychological Perspectives
*Gender Identity in Transition
*Special Topics: Psychology of Women
*Issues in Domestic Violence
*Women and Crime

==== Arts and Humanities ====

'''Suggested courses for this track:'''

*Women in Art
*Identity and Culture: Gender and Media in Global Perspective
*Women and Gender in Film
*Honors Seminar in Literature: Emily Dickinson 
*Honors Seminar in Literature: Austen and the Brontes
*Film and Literature: Austen/Shelley/Hollywood
*Special Topics: Literature of Women
*Women Writers
*Voices of Medieval Women
*Topics in British Literature: Virginia Woolf
*Topics in World Literature: African-American Writers
*Topics in World Literature: Black Women Writers
*Topics in British Literature: Unfortunate Women
*Women in 20th Century U.S. History
*Comparative History of the Modern Family: 1750-Present 
*Historical Themes: Women in Ancient Greece
*Historical Themes: Modern European Women
*Historical Themes: African-American Women in History
*Women in Western Music
*Feminist Philosophy
*Philosophical Topics: Simone de Beauvoir 
*Hispanic Popular Cultures: Mexican Cultural Symbols: La Malinche, La Virgen de Guadalupe and La Llorona
*Special Topics: Latin American Women Writers

==== Interdisciplinary ====

'''Suggested courses for this track:'''

*Themes and Issues in Women’s Studies
*Women’s Culture and Creativity 
*Women’s Words, Women’s Lives
*Women, Work and Family
*Women and Medicine
*Women and Sexuality
*Women and Aging
*Topics in Women’s Studies: Gender, Development and Globalization 
*Topics in Women’s Studies: Women, the Environment and Health
*Seminar in Women’s Studies
*Directed Readings for Honors in Women’s Studies
*Directed Readings for Women’s Studies
*Internship in Women’s Studies
*Honors Thesis

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this department, you can list your name here (this can help small departments grow and the participants communicate better).
* [[User:Mehmetaergun|mehmetaergun]] 16:32, 19 October 2006 (UTC)

===Books===

===Wikipedia===
* [[w:Women's studies|Women's Studies]]
* [[w:First-wave feminism|First-wave feminism]]
* [[w:Second-wave feminism|Second-wave feminism]]
* [[w:Third-wave feminism|Third-wave feminism]]
* [[w:Feminist studies|Feminist studies]]
* [[w:Feminism|Feminism]]
* [[w:Men's studies|Men's studies]]
* [[w:Radical feminism|Radical feminism]]
* [[w:Psychoanalytic feminism|Psychoanalytic feminism]]
* [[w:Anarcha-feminism|Anarcha-feminism]]
* [[w:New feminism|New feminism]]
* [[w:Domestic violence|Domestic violence]]
* [[w:Equal pay for women|Equal pay for women]]
* [[w:Equal Rights Amendment|Equal Rights Amendment]]
* [[w:Feminist history in the United States|Feminist history in the United States]]
* [[w:Feminist history in the United Kingdom|Feminist history in the United Kingdom]]
* [[w:Feminism in Poland|Feminist history in Poland]]
* [[w:Feminist history in Latin America|Feminist history in Latin America]]
* [[w:Feminist theology|Feminist theology]]
* [[w:Gender studies|Gender studies]]
* [[w:Gender-neutral language|Gender-neutral language]]
* [[w:History of feminism|History of feminism]]
* [[w:Islamic feminism|Islamic feminism]]
* [[w:Lesbian feminism|Lesbian feminism]]
*'''[[w:List of feminism topics|List of feminism topics]]'''
*'''[[w:List of notable feminists|List of notable feminists]]'''
* [[w:Role of women in Judaism|Role of women in Judaism]]
* [[w:Suffragettes|Suffragettes]]
* [[w:Women's cinema|Women's cinema]]
* [[w:Women's Environment &amp; Development Organization|Women's Environment &amp; Development Organization]]
* [[w:Women's music|Women's music]]
* [[w:Sex/gender distinction|Sex/gender distinction]]

===Wikibooks===
* [[b:Feminism|Feminism]] (see the [[b:Feminism/Contents|contents]])
* [[b:Introduction to Sociology/Gender|Introduction to the Sociology of Gender]]
* [[b:Literary Criticism/Contents/Feminist Literary Criticism|Feminist Literary Criticism]]

==External resources==
* [http://bailiwick.lib.uiowa.edu/wstudies/ Women's Studies Resources at the University of Iowa] - This site, in continuous operation since 1996, features the following annotated links pages: Activism, Art, Communication and Media, Development - WID, Feminist Theory, General or Mixed Indexes &amp; Searches, History, Literature, Music and Sports. Not connected to the U. Iowa Women's Studies Department.
* [http://research.umbc.edu/~korenman/wmst/wmst-l index.html Women's Studies Email List (WMST-L)] - WMST-L is an international electronic forum for people involved in Women's Studies as teachers, researchers, librarians, and/or program administrators. It offers a rapid and cost-free way for participants to ask questions and exchange information about the academic side of Women's Studies: current research, teaching strategies, useful texts and films, online resources, innovative courses, building Women's Studies majors, minors, and graduate programs, and other academic issues. WMST-L also welcomes announcements about relevant conferences, calls for papers, job opportunities, publications, and the like. 
* [http://research.umbc.edu/~korenman/wmst/wmsttoc.html WMST-L's online collection of files] related to Women's Studies.
* [http://research.umbc.edu/~korenman/wmst/links.html Women's Studies/Women's Issues Resource Sites] - A frequently-updated, annotated, selective set of more than 700 links to sites offering resources and information about women's studies and/or women's issues. Includes 16 topical subsections, such as Activism, Arts and Humanities, Health, Cyberculture and Internet Information, Science and Technology, Sexuality and Sexual Orientation, Women of Color, and more.
* [http://research.umbc.edu/~korenman/wmst/programs.html Women's Studies Programs Worldwide] - Approximately 700 listings in the US and around the world. Annotations identify programs offering graduate degrees or certificates. Frequently updated.
* [http://research.umbc.edu/~korenman/wmst/forums.html Gender-Related Electronic Forums] - A frequently-updated listing and description of more than 600 e-mail lists that focus on women- or gender-related issues. Includes 16 topical subsections, such as Activism, Arts and Humanities, Business, Health, Motherhood, Religion and Spirituality, Science and Technology, Sports and Recreation, Sexuality and Sexual Orientation, Women of Color, and more.
* [http://www.umbc.edu/cwit/ Center for Women and Information Technology] - ABCNews.com has called this site &quot;the best resource on women and technology on the Web.&quot; The site offers information, news, resources, and initiatives that address women's use of information technology [IT] and their under-representation in the IT workforce. Includes a very large collection of [http://www.umbc.edu/cwit/syllabi.html web-based women- and gender-related syllabi] and information about [http://www.umbc.edu/cwit/financial aid.html financial aid for women] including women not in IT.
* [http://linuxchix.org/ Linux Chix] - LinuxChix is a community for women who like Linux, and for women and men who want to support women in computing.
* [http://women.debian.org/ Debian Women], founded in May 2004, seeks to balance and diversify the [[w:Debian|Debian]] Project (a major [[w:Linux|Linux]] distribution) by actively engaging with interested women and encouraging them to become more involved with Debian.
* [http://ubuntu-women.org/ Ubuntu Women] is a team functioning under [[w:Ubuntu|Ubuntu]] (a major [[w:Linux|Linux]] distribution) to provide a platform and encouragement for women to contribute to Ubuntu-Linux, a Debian based free and open-source GNU/Linux software.
* [[Heterosexuality Questionnaire]] that makes one think about how heterosexuality is taken for granted.
* [http://www.wikigender.org Wikigender]: a wiki about gender equality and women's rights 
* [http://nebraskavolleyballschedule.us/ Women's Volleyball] Resources - Women's volleyball has become one of the fastest-growing and most inspiring sports worldwide, providing opportunities for female athletes to compete, develop leadership skills, and build confidence through teamwork. Women's volleyball communities offer schedules, player updates, team news, training resources, and information about major competitions. Programs like Nebraska Women's Volleyball continue to inspire fans with dedication, athletic excellence, and a commitment to growing the sport.

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'''Part of  [[:Category:Social Sciences|Social Sciences]]'''

==Introduction and Descripton==

'''Welcome to Wikiversity's Department of Women's Studies'''

Hello! The Wikiversity Department of Women's Studies has just been founded. If you are knowledgable about any of the topics listed below, or about other Women's Studies topics, your help would be extremely appreciated if you make a contribution.

Please do not delete learning projects. Instead, try to add a project if you would like it to be offered or if you would like to offer it.

==Learning Resources==
{{Template:Courses}}

===Core Lessons===

* [[Introduction to Wiki]] '''AND'''
* [[/Women in Perspective/]] '''OR'''
* [[/International Perspectives on Women/]] '''AND'''
* [[/Feminist Theory/]]

===Tracks===

'''Please Choose Any Section Below:'''

==== Multiculturalism and Diversity ====

* [[/International Perspectives on Women/]]
* [[/Women of Color/]]
* [[/Trans Perspectives in the Early 21st Century/]]
* [[/Gender in Cross-Cultural Perspective/]]
* [[/Latinas in the Americas/]]
* [[/World Literature: African Women Writers/]]
* [[/African-American Women Writers/]]
* [[Gays and Lesbians in U.S. History]]
* [[Queer Studies]]
* [[/Life Histories of African Women/]]
* [[/Race, Class and Gender/]]
* [[/Women in World Religions/]]
* [[/Psychology of Lesbian Culture/]]
* [[/Sociology of Race, Class and Gender/]]

=== Suggested tracks ===

==== Nature; Science ====

'''Suggested courses for this track:'''

*Reproductive Technologies and the Future of Motherhood
*Biology of Women
*Economic Issues of Gender
*Women’s Health
*Perspectives on Rape and Sexual Assault
*Women, Gender and Science
*Women, Society and Radiation
*The American Woman in Sport
*Special Topics: Women and Law
*Sex Differences: Psychological Perspectives
*Gender Identity in Transition
*Special Topics: Psychology of Women
*Issues in Domestic Violence
*Women and Crime

==== Arts and Humanities ====

'''Suggested courses for this track:'''

*Women in Art
*Identity and Culture: Gender and Media in Global Perspective
*Women and Gender in Film
*Honors Seminar in Literature: Emily Dickinson 
*Honors Seminar in Literature: Austen and the Brontes
*Film and Literature: Austen/Shelley/Hollywood
*Special Topics: Literature of Women
*Women Writers
*Voices of Medieval Women
*Topics in British Literature: Virginia Woolf
*Topics in World Literature: African-American Writers
*Topics in World Literature: Black Women Writers
*Topics in British Literature: Unfortunate Women
*Women in 20th Century U.S. History
*Comparative History of the Modern Family: 1750-Present 
*Historical Themes: Women in Ancient Greece
*Historical Themes: Modern European Women
*Historical Themes: African-American Women in History
*Women in Western Music
*Feminist Philosophy
*Philosophical Topics: Simone de Beauvoir 
*Hispanic Popular Cultures: Mexican Cultural Symbols: La Malinche, La Virgen de Guadalupe and La Llorona
*Special Topics: Latin American Women Writers

==== Interdisciplinary ====

'''Suggested courses for this track:'''

*Themes and Issues in Women’s Studies
*Women’s Culture and Creativity 
*Women’s Words, Women’s Lives
*Women, Work and Family
*Women and Medicine
*Women and Sexuality
*Women and Aging
*Topics in Women’s Studies: Gender, Development and Globalization 
*Topics in Women’s Studies: Women, the Environment and Health
*Seminar in Women’s Studies
*Directed Readings for Honors in Women’s Studies
*Directed Readings for Women’s Studies
*Internship in Women’s Studies
*Honors Thesis

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this department, you can list your name here (this can help small departments grow and the participants communicate better).
* [[User:Mehmetaergun|mehmetaergun]] 16:32, 19 October 2006 (UTC)

===Books===

===Wikipedia===
* [[w:Women's studies|Women's Studies]]
* [[w:First-wave feminism|First-wave feminism]]
* [[w:Second-wave feminism|Second-wave feminism]]
* [[w:Third-wave feminism|Third-wave feminism]]
* [[w:Feminist studies|Feminist studies]]
* [[w:Feminism|Feminism]]
* [[w:Men's studies|Men's studies]]
* [[w:Radical feminism|Radical feminism]]
* [[w:Psychoanalytic feminism|Psychoanalytic feminism]]
* [[w:Anarcha-feminism|Anarcha-feminism]]
* [[w:New feminism|New feminism]]
* [[w:Domestic violence|Domestic violence]]
* [[w:Equal pay for women|Equal pay for women]]
* [[w:Equal Rights Amendment|Equal Rights Amendment]]
* [[w:Feminist history in the United States|Feminist history in the United States]]
* [[w:Feminist history in the United Kingdom|Feminist history in the United Kingdom]]
* [[w:Feminism in Poland|Feminist history in Poland]]
* [[w:Feminist history in Latin America|Feminist history in Latin America]]
* [[w:Feminist theology|Feminist theology]]
* [[w:Gender studies|Gender studies]]
* [[w:Gender-neutral language|Gender-neutral language]]
* [[w:History of feminism|History of feminism]]
* [[w:Islamic feminism|Islamic feminism]]
* [[w:Lesbian feminism|Lesbian feminism]]
*'''[[w:List of feminism topics|List of feminism topics]]'''
*'''[[w:List of notable feminists|List of notable feminists]]'''
* [[w:Role of women in Judaism|Role of women in Judaism]]
* [[w:Suffragettes|Suffragettes]]
* [[w:Women's cinema|Women's cinema]]
* [[w:Women's Environment &amp; Development Organization|Women's Environment &amp; Development Organization]]
* [[w:Women's music|Women's music]]
* [[w:Sex/gender distinction|Sex/gender distinction]]

===Wikibooks===
* [[b:Feminism|Feminism]] (see the [[b:Feminism/Contents|contents]])
* [[b:Introduction to Sociology/Gender|Introduction to the Sociology of Gender]]
* [[b:Literary Criticism/Contents/Feminist Literary Criticism|Feminist Literary Criticism]]

==External resources==
* [http://bailiwick.lib.uiowa.edu/wstudies/ Women's Studies Resources at the University of Iowa] - This site, in continuous operation since 1996, features the following annotated links pages: Activism, Art, Communication and Media, Development - WID, Feminist Theory, General or Mixed Indexes &amp; Searches, History, Literature, Music and Sports. Not connected to the U. Iowa Women's Studies Department.
* [http://research.umbc.edu/~korenman/wmst/wmst-l index.html Women's Studies Email List (WMST-L)] - WMST-L is an international electronic forum for people involved in Women's Studies as teachers, researchers, librarians, and/or program administrators. It offers a rapid and cost-free way for participants to ask questions and exchange information about the academic side of Women's Studies: current research, teaching strategies, useful texts and films, online resources, innovative courses, building Women's Studies majors, minors, and graduate programs, and other academic issues. WMST-L also welcomes announcements about relevant conferences, calls for papers, job opportunities, publications, and the like. 
* [http://research.umbc.edu/~korenman/wmst/wmsttoc.html WMST-L's online collection of files] related to Women's Studies.
* [http://research.umbc.edu/~korenman/wmst/links.html Women's Studies/Women's Issues Resource Sites] - A frequently-updated, annotated, selective set of more than 700 links to sites offering resources and information about women's studies and/or women's issues. Includes 16 topical subsections, such as Activism, Arts and Humanities, Health, Cyberculture and Internet Information, Science and Technology, Sexuality and Sexual Orientation, Women of Color, and more.
* [http://research.umbc.edu/~korenman/wmst/programs.html Women's Studies Programs Worldwide] - Approximately 700 listings in the US and around the world. Annotations identify programs offering graduate degrees or certificates. Frequently updated.
* [http://research.umbc.edu/~korenman/wmst/forums.html Gender-Related Electronic Forums] - A frequently-updated listing and description of more than 600 e-mail lists that focus on women- or gender-related issues. Includes 16 topical subsections, such as Activism, Arts and Humanities, Business, Health, Motherhood, Religion and Spirituality, Science and Technology, Sports and Recreation, Sexuality and Sexual Orientation, Women of Color, and more.
* [http://www.umbc.edu/cwit/ Center for Women and Information Technology] - ABCNews.com has called this site &quot;the best resource on women and technology on the Web.&quot; The site offers information, news, resources, and initiatives that address women's use of information technology [IT] and their under-representation in the IT workforce. Includes a very large collection of [http://www.umbc.edu/cwit/syllabi.html web-based women- and gender-related syllabi] and information about [http://www.umbc.edu/cwit/financial aid.html financial aid for women] including women not in IT.
* [http://linuxchix.org/ Linux Chix] - LinuxChix is a community for women who like Linux, and for women and men who want to support women in computing.
* [http://women.debian.org/ Debian Women], founded in May 2004, seeks to balance and diversify the [[w:Debian|Debian]] Project (a major [[w:Linux|Linux]] distribution) by actively engaging with interested women and encouraging them to become more involved with Debian.
* [http://ubuntu-women.org/ Ubuntu Women] is a team functioning under [[w:Ubuntu|Ubuntu]] (a major [[w:Linux|Linux]] distribution) to provide a platform and encouragement for women to contribute to Ubuntu-Linux, a Debian based free and open-source GNU/Linux software.
* [[Heterosexuality Questionnaire]] that makes one think about how heterosexuality is taken for granted.
* [http://www.wikigender.org Wikigender]: a wiki about gender equality and women's rights

[[Category:Women's studies]]</text>
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        <id>3105359</id>
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      <text bytes="4904" sha1="6qnjd34peuhyqmlzh6czxgw0mqqofw1" xml:space="preserve">This is the '''Photoproject''' for learning about [[photography]] of all types. Go out and take pictures! Bring them here for discussion.

==Scope==

==Projects==
===Time-lapse===
I want to try some [[w:Time-lapse|time-lapse]] photography where a picture is taken every day. I'm worried about how to make sure that each picture is taken in the same way. Is it possible without a tripod? --[[User:JWSchmidt|JWSchmidt]] 03:50, 9 July 2007 (UTC)

:If you want an identical frame, you really should use a tripod, if possible. Otherwise, It's going to be very hard (if impossible) to get it right each time. If it's not possible to use a tripod, you could use an object that is fixed on a particular spot, and take the photograph resting on exactly the same spot on this object every day, noting where the sides of the frame (ie picture) are marked in the scene. This will be made easier by a camera with a clear (and preferably large) viewfinder - traditional, film-based compact cameras will not be good for this - an [[w:Single-lens reflex camera|SLR]] would be better, and a decent digital compact would probably be fine too. [[User:Cormaggio|Cormaggio]] &lt;sup&gt;&lt;small&gt;[[User talk:Cormaggio|talk]]&lt;/small&gt;&lt;/sup&gt; 22:21, 9 July 2007 (UTC)
:Feel free to show us how it turns out.[[User:Elatanatari|Elatanatari]] 20:52, 12 July 2007 (UTC)

Also there is a way to make sure that the pictures are aligned. Take your camera and take pictures everyday at a relative spot like on something fixed. Then you can upload the pictures to photoshop and put it in layers, there is an option that will try to auto align the layers for you. Then you will have to crop it and take the sides off because some pictures might be off a little so photoshop will move it to the side a bit. After that you then can proceed to make a video out of it. I have already done something similar and uploaded to youtube. the link is http://www.youtube.com/watch?v=s64qF0v2AG0 Also I am doing a similar project and I have a forum for it. --Rlin06

 Pakistan Muslim League Zia Khalida ever first in history of Pakistan with extremely new ideology, with honored of magnificent historical background.
   Pakistan Muslim league Zia Khalida  has significant performances held behind the screen, follower of the Quaid E Azam Muhammad Ali Jinnah , PML-ZK believe on the four provinces map as been made by Quaid E Zam Muhammad Ali Jinnah, strong Defence to make secure the country,all schools colleges &amp; universities must not higher fees for students for admission in any education institute, agriculture sectors must be safe by town ships , reduce pollution and expansion of the Trees (jungles) inside every province of Pakistan , economy to make it control on normal or average line         
PML-ZK believe Strong Pakistan Strong Economy &amp; wealthy societies . ban Ammunition culture, reformation the recreations on Infrastructure of of cities of Pakistan.

==Issues==
If you have an issue you would like to raise about any aspect of photography, please add it here.


===New Camera===

I'm getting a new camera. I want an SLR, but I'm broke. What should I get? Where should I look?[[User:Elatanatari|Elatanatari]] 03:42, 11 May 2008 (UTC)
:Do you want a digital or film-based SLR? You can pick up good second-hand film-based cameras quite cheap (though you'll have to pay for subsequent processing of film and prints). Where are you based? [[User:Cormaggio|Cormaggio]] &lt;sup&gt;&lt;small&gt;[[User talk:Cormaggio|talk]]&lt;/small&gt;&lt;/sup&gt; 11:32, 11 May 2008 (UTC)
::Digital, I already have a great film SLR, but the processing can get pricy. Detroit.[[User:Elatanatari|Elatanatari]] 15:55, 11 May 2008 (UTC)
::Also what do you think of bridge cameras?--[[User:Elatanatari|Elatanatari]] 00:20, 16 June 2008 (UTC)
:::Well, I suppose if you're broke, then second-hand is going to be a real option. There are always people who get a middle-range camera and who subsequently want to upgrade to a pro-end camera, so you might be able to get something decent on, say, eBay. For specific digital camera reviews, [http://www.dpreview.com/ this] is a good website. I think it's always worth thinking ahead when buying a camera - will it have enough resolution to produce decent-sized prints? What will I be using it for (will I need particular lenses or other additionals in the future)? I think this is the major drawback of &quot;bridge&quot; or any other non-SLR - you can't add any lenses to it down the line. I'd recommend a Canon or Nikon - they have a great range of lenses - but other major manufacturers, like Pentax, Minolta, Olympus, etc are pretty good too. Let me know what you find out, and I'll try to help. [[User:Cormaggio|Cormaggio]] &lt;sup&gt;&lt;small&gt;[[User talk:Cormaggio|talk]]&lt;/small&gt;&lt;/sup&gt; 09:05, 25 June 2008 (UTC)

==See also==
* [[Topic:Photography|The Department of Photography]]

[[Category:Photography]]
[[Category:Learning projects]]
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  <page>
    <title>Firefox</title>
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      <text bytes="11818" sha1="mbe0vn7nfrn6gioa3nud08h66z3ofv1" xml:space="preserve">==Setup==
* [https://support.mozilla.org/en-US/kb/make-firefox-your-default-browser How to make Firefox the default browser]

==Shortcuts==
* CTRL+W closes a tab
* CTRL+T opens a new tab
* CTRL+SHIFT+T reopens the last closed tab
* CTRL+↹Tabulator; CTRL+Shift+↹Tabulator; CTRL+PgUp/PgDn: switch through tabs.
* CTRL+Shift+PgUp/PgDn: move opened tab left or right respectively.
* F6 or Ctrl+L : Gets you right up into the Address/URL bar.
* F5 : Reload the page.
* CTRL+F5: Clear cache for the current page and reload the page.
* / or Ctrl+F: Search/find text on page
* Ctrl+K : Takes you to the Firefox search box.
* Ctrl+U : View the page’s source code.
* F10 : Access top menu (File, Edit, View, History, …).
* F11 : View the page in full-screen mode.
* F12 : Access developer tools such as page inspector.
* Ctrl+W : Closes the active tab.
* Ctrl+&lt;kbd&gt;=&lt;/kbd&gt; : Resets font size.
* Ctrl+&lt;kbd&gt;+&lt;/kbd&gt; : Increases font size.
* Ctrl+&lt;kbd&gt;-&lt;/kbd&gt; : Decreases font size.
* Ctrl+S : Save page
* Ctrl+Shift+I, F12: Open web development tools.

== Downloading ==
There are four main ways to download an internet resource:

* &lt;kbd&gt;Ctrl+S&lt;/kbd&gt; – Download currently viewed page. 

In the file saving dialogue, &quot;Web Page, complete&quot; takes the loaded page from memory and stores resources such as multimedia and style sheets and JavaScripts as individual files inside a &lt;code&gt;_files&lt;/code&gt; suffix directory. &quot;Web Page, HTML only&quot; saves the verbatim page source code downloaded from the server.

* &quot;Save Link as&quot; from right-click context menu (saves verbatim page source code)
* Dragging the URL or tab into the Download arrow in the tool bar.
* Pasting an URL from the clipboard into the opened download history (in the &quot;Library&quot; window or &lt;code&gt;about:downloads&lt;/code&gt;).

Additionally, the page source code can be saved by copying it from the web development tools and pasted into a text editor.

== Profiles == 
User profiles allow separating configuration, extensions, bookmarks, and history within Firefox. By default, there is one profile. The profile manager allows creating additional profiles. It can be opened by launching the Firefox process through a command line with the &lt;code&gt;-profilemanager&lt;/code&gt; parameter, without spaces.

Profile folders are typically located under &lt;code&gt;%AppData%\Mozilla\Firefox\&lt;/code&gt; on Microsoft Windows,&lt;code&gt;~/Library/Application Support/Firefox/&lt;/code&gt; on MacOS, and &lt;code&gt;~/.mozilla/firefox/&lt;/code&gt; on Linux and BSD. The profile manager allows manually specifying custom profile directories.&lt;ref&gt;[https://support.mozilla.org/en-US/kb/profile-manager-create-remove-switch-firefox-profiles?redirectslug=profile-manager-create-and-remove-firefox-profiles&amp;redirectlocale=en-US ''Profile Manager - Create, remove or switch Firefox profiles'' – Mozilla Support (knowledge base)]&lt;/ref&gt;

Relevant files in the profile folder are &lt;code&gt;places.sqlite&lt;/code&gt; for browsing history and download history and bookmarks, &lt;code&gt;cookies.sqlite&lt;/code&gt;, &lt;code&gt;prefs.js&lt;/code&gt; for user preferences, and &lt;code&gt;containers.json&lt;/code&gt; for configuration of multi-account containers. Sessions are stored in &lt;code&gt;sessionstore.jsonlz4&lt;/code&gt; when closing the browser (before Firefox 56 of 2017, it was stored as plain JSON into &lt;code&gt;sessionstore.json&lt;/code&gt;), and backed up to &lt;code&gt;sessionstore-backups/recovery.jsonlz4&lt;/code&gt; and &lt;code&gt;sessionstore-backups/recovery.baklz4&lt;/code&gt; while browsing for recovery in case of an unexpected process termination. The previous session is backed up to &lt;code&gt;sessionstore-backups/previous.jsonlz4&lt;/code&gt;. &lt;ref&gt;[https://support.mozilla.org/en-US/kb/profiles-where-firefox-stores-user-data Profiles - Where Firefox stores your bookmarks, passwords and other user data – Mozilla Support]&lt;/ref&gt;

== Advanced configuration ==
On the desktop edition (and earlier mobile versions), the &lt;code&gt;about:config&lt;/code&gt; page allows fine-tuning the browser. The search bar at the top facilitates finding properties.

Notable properties are:

* &lt;code&gt;browser.urlbar.trimURLs&lt;/code&gt; – may hide the protocol from the URL bar. Deactivate to always show the full URL including the protocol. Activated by default.
* &lt;code&gt;browser.tabs.insertAfterCurrent&lt;/code&gt; – open new tabs next to the currently active tab instead of at the end of the tab list.
* &lt;code&gt;javascript.enabled&lt;/code&gt; – deactivating usually increases speed and decreases CPU usage of [[:w:Progressive enhancement|progressively enhancing]] sites, but features implemented using JavaScript such as [[:mw:Extension:WikiEditor|the toolbar of the wikitext form]] and JavaScript-based sites such as Twitter web app will not work. Activated by default.
* &lt;code&gt;browser.backspace_action&lt;/code&gt; – this parameter adjusts the function of the backspace key if no text input field is active. &lt;code&gt;0&lt;/code&gt;: navigate to last page, &lt;code&gt;1&lt;/code&gt;: scroll up, if ''Shift'' key is held then scroll down. &lt;code&gt;2&lt;/code&gt;: deactivate.
* &lt;code&gt;dom.event.contextmenu.enabled&lt;/code&gt; – allows JavaScript to interfere with right-click context menu. Deactivating might interfere with some sites' functionality. Activated by default.
* &lt;code&gt;dom.event.clipboardevents.enabled&lt;/code&gt; – allows JavaScript to evaluate on-page text selection. Deactivating might interfere with some sites' functionality. Activated by default.
* &lt;code&gt;browser.download.autohideButton&lt;/code&gt; – hide download button if no downloads in current session. Activated by default.
* &lt;code&gt;browser.download.alwaysOpenPanel&lt;/code&gt; – Open list of recent downloads after a download finished.
* &lt;code&gt;browser.download.lastDir&lt;/code&gt;, &lt;code&gt;browser.open.lastDir&lt;/code&gt; – last directory to which a file was downloaded or from which a file was opened through the file picker dialogue.
* &lt;code&gt;browser.startup.homepage&lt;/code&gt; – page displayed when starting a new session (if not restoring the previous session), and when pressing the button with the house icon.
* &lt;code&gt;dom.ipc.processCount.web&lt;/code&gt; – number of processes across which web content is distributed. Higher count increases performance but consumes more memory, thus recommended for computers with much RAM. Four by default. 
* &lt;code&gt;general.autoScroll&lt;/code&gt; – enable scrolling using middle mouse button press.
* &lt;code&gt;devtools.chrome.enabled&lt;/code&gt; – enable write access to browser console, accessed using Ctrl+Shift+J.
* &lt;code&gt;browser.cache.disk.capacity&lt;/code&gt;, &lt;code&gt;browser.cache.offline.capacity&lt;/code&gt; disk cache capacity, mainly used for static resources to save bandwidth. Difference between values unclear yet. – Values in Kilobytes.
* &lt;code&gt;places.history.expiration.transient_current_max_pages&lt;/code&gt;: number of entries retained in the browsing history database (&lt;code&gt;places.sqlite&lt;/code&gt; file). According to a moderator of Mozilla's support forum, the value is read-only by default but can be adjusted through the &lt;code&gt;places.history.expiration.max_pages&lt;/code&gt; property, which has to be added manually.&lt;ref&gt;[https://support.mozilla.org/en-US/questions/1275209#answer-1274292 Answer to ''FF 71. Cannot change value in setting places.history.expiration.transient_current_max_pages'' | Firefox Support Forum | Mozilla Support (December 17, 2019)]&lt;/ref&gt;
* &lt;code&gt;accessibility.blockautorefresh&lt;/code&gt; – blocks &quot;meta refreshes&quot; and &quot;meta redirects&quot;, meaning refreshes and redirects using the &lt;code&gt;meta http-equiv&lt;/code&gt; tag, as well as redirects instructed by the &lt;code&gt;refresh&lt;/code&gt; [[:w:HTTP header|HTTP header]]. However, it does not block JavaScript-based redirects and refreshes, i. e. &lt;code&gt;document.location.href&lt;/code&gt;.&lt;ref&gt;[https://support.mozilla.org/en-US/questions/1327430 ''Enable/disable automatic redirect. | Firefox Support Forum | Mozilla Support'' (2/26/21)]&lt;/ref&gt;
* &lt;code&gt;security.csp.enable&lt;/code&gt; – Claimed by Mozilla to increase security against &quot;cross-site scripting&quot; (XSS) attacks.&lt;ref&gt;[https://developer.mozilla.org/en-US/docs/Web/HTTP/CSP Content Security Policy (CSP) - HTTP] from Mozilla Developers Network&lt;/ref&gt; User reports suggest it interferes with the function of [[:w:JavaScript bookmarklet|bookmarklet]]s.&lt;ref&gt;[https://superuser.com/questions/586063/how-to-disable-csp-in-firefox-for-just-bookmarklets ''How to disable CSP in Firefox for just bookmarklets?'' – Super User]&lt;/ref&gt;
* &lt;code&gt;permissions.default.image&lt;/code&gt; – 1: Load images as usual (default); 2: Do not load images (usually for testing purposes or to save bandwidth);  3: Only load images from the same domain.&lt;ref&gt;https://support.mozilla.org/en-US/questions/981640&lt;/ref&gt;
* &lt;code&gt;media.videocontrols.picture-in-picture.video-toggle.enabled&lt;/code&gt; (or similar): show the picture-in-picture button in video players.
* &lt;code&gt;security.dialog_enable_delay&lt;/code&gt; – delay in milliseconds until the confirmation button in some dialogue boxes like downloading an executable file or installing an extension gets unlocked. Until then, it is [[:w:Grayed_out|&quot;grayed out&quot;]]. Default value: 1000 (1 second).&lt;ref&gt;[http://kb.mozillazine.org/Disable_extension_install_delay_-_Firefox ''Disable extension install delay - Firefox - MozillaZine Knowledge Base'']&lt;/ref&gt;
* &lt;code&gt;browser.cache.disk.enable&lt;/code&gt; – enable saving sites' resources locally to speed up loading.
* &lt;code&gt;browser.cache.disk.capacity&lt;/code&gt; – disk cache size in kilobytes.
* &lt;code&gt;browser.urlbar.update2.engineAliasRefresh&lt;/code&gt; – allow adding custom search engines.&lt;ref&gt;[https://superuser.com/questions/7327/how-to-add-a-custom-search-engine-to-firefox/1756774#1756774 How to add a custom search engine to Firefox? - Super User]&lt;/ref&gt;
* &lt;code&gt;accessibility.typeaheadfind.matchesCountLimit&lt;/code&gt; – Sets the limit for counting text matches when searching text on the current page. A high limit can slow down the browser on long pages. The default is 1000.
* &lt;code&gt;browser.privatebrowsing.forceMediaMemoryCache&lt;/code&gt; – While in [[:w:private_browsing|private browsing]], video and audio content is only cached into the computer's RAM, not the data storage. No such option exists for normal browsing as of early 2025.
* &lt;code&gt;browser.menu.showViewImageInfo&lt;/code&gt; – Show the &quot;View Image Info&quot; option in the context menu that appears when right-clicking a picture.
* &lt;code&gt;intl.date_time.pattern_override.date_short&lt;/code&gt; – Can be used to set a custom date and time format shown in various places such as the browsing history. For example &lt;code&gt;YYYY-MM-dd&lt;/code&gt; for ISO 8601.
* &lt;code&gt;full-screen-api.warning.timeout&lt;/code&gt; – How long (in milliseconds) the &quot;[domain name] is now full screen&quot; message is shown after switching to full screen mode. Set to 0 to completely deactivate it.

== System pages ==
* &lt;code&gt;about:memory&lt;/code&gt; – internal process viewer.
* &lt;code&gt;about:processes&lt;/code&gt; – newer internal process manager. Introduced with Firefox 67 in early 2019.&lt;ref&gt;https://www.ghacks.net/2019/03/01/firefox-67-automatically-unload-unused-tabs-to-improve-memory/&lt;/ref&gt; Can be used to find tabs that are playing media in the background.
* &lt;code&gt;about:performance&lt;/code&gt; – similar to &lt;code&gt;about:processes&lt;/code&gt;, but with more beginner-friendly worded user interface, such as &quot;Energy impact&quot; instead of &quot;CPU&quot;. Introduced in late 2018.&lt;ref&gt;[https://www.ghacks.net/2018/10/11/this-is-firefoxs-upcoming-aboutperformance-page-huge-improvements/ This is Firefox's upcoming about:performance page (huge improvements) – Martin Brinkmann, 2018-10-11]&lt;/ref&gt;

==See also==
* [[w:Iceweasel|Iceweasel]] browser
* [[devmo:|Firefox Development Wiki]]
* [[MozillaZineKB:|Mozilla Knowledge-Base Wiki]]
* [[links (browser)]], [[lynx (browser)]], [[Elinks (browser)]] command line browsers
* [[CURL (software)|cURL]]

==External links==
* [http://www.qedoc.org/en/index.php?title=Mozilla_Firefox Mozilla Firefox] - Interactive learning quiz at Qedoc.org

[[Category:Firefox]]
[[Category:Web browsers]]</text>
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  <page>
    <title>Sport</title>
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      <text bytes="1297" sha1="0qvcyz31i06b46sd14xym9n66zm106p" xml:space="preserve">[[File:Judo Morote Seoi Nage by Bryan small.gif|thumb|270px|A shoulder throw at Judo rank graduation, [[Japan]].]]{{commons}}'''Sport''' is commonly defined as a sort of game that requires physical activity and involves a degree of competition as, for example, baseball, [[soccer]], bowling, or basketball.  Many adolescents and adults play sports with their friends. Sportsmen need coaches to teach or train teams or individuals how to do better. Sports can be played indoors or outdoors.

Some people like to watch other people play sports. Those who like to watch people playing sports are called ''fans''. While some fans watch sports on television, others actually go to places where people play sports to watch them in person. These fans are called ''spectators''.

At present, there is a great variety of sports available, for example:[[File:FIFA World Cup 2010 Argentina vs Germany - Thomas Müller opening goal.gif|thumb|270px|FIFA World Cup 2010 Argentina vs Germany.]]
&lt;div style=&quot;-moz-column-count:2; column-count:2;&quot;&gt;
* [[Acrobatics]]
* [[Bodybuilding]]
* [[Cricket]]
* [[Gymnastics]]
* [[Rhythmic gymnastics]]
* [[Diving]]
* [[Swimming]]
* [[Football]]
* [[Volleyball]]
* [[Association football]]
&lt;/div&gt;

{{Physical exercising}}
[[Category:Sports| ]]
[[Category:Physical exercising]]</text>
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    <title>Understanding Arithmetic Circuits</title>
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        <username>Young1lim</username>
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      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260803.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20260803.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

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    <title>Complex analysis in plain view</title>
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      <text bytes="5296" sha1="9oak62n97925eck0xqqd74dwp5pptwz" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260803.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

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    <title>Universal Bibliography</title>
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      <text bytes="44362" sha1="fbvcvc3nqnftqtrx8gqw8xfg0pbnhrp" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; Dictionary of National Biography
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*Playthings: The National Magazine of the Toy Trade. [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. Geyer-McAllister Publications.

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Cinema==
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; Dictionary of National Biography
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*Playthings: The National Magazine of the Toy Trade. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. Geyer-McAllister Publications.

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Cinema==
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; Dictionary of National Biography
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*Playthings: The National Magazine of the Toy Trade. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. Geyer-McAllister Publications. Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Cinema==
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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[[Category:Research]]</text>
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; Dictionary of National Biography
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*Playthings: The National Magazine of the Toy Trade. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Cinema==
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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      <text bytes="44536" sha1="lfuwzkx2v5bto5z0eglpxeyu020gbh8" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; Dictionary of National Biography
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Cinema==
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
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See also [[Universal Bibliography/Bibliography|Bibliography]]

This part of the [[Universal Bibliography]] is a bibliography of literature.

See [[s:Category:Literature]], [[s:Category:History of literature]], [[w:Bibliography of encyclopedias: literature]] and [[w:Category:Works about literature]]

==World==
*Cassell's Encyclopaedia of World Literature. 1953. 2nd Ed: 1973. vol 1 (histories and general articles): [https://books.google.co.uk/books?id=soIYAAAAIAAJ]. vol 3: [https://books.google.co.uk/books?id=AIEYAAAAIAAJ]. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA197#v=onepage&amp;q&amp;f=false].

*Damrosch. What is World Literature? 2003. [https://books.google.co.uk/books?id=yY-17mtp9R8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series:
*Edmund Gosse (ed). Literatures of the World. D Appleton and Co [https://en.m.wikisource.org/wiki/Page:A_history_of_Bohemian_literature.pdf/444]

==English==
*Bateson. Cambridge Bibliography of English Literature. 1940. Watson, Supplement 1957.
*Concise Cambridge Bibliography of English Literature
*New Cambridge Bibliography of English Literature
*Annual Bibliography of English Language and Literature. Cambridge University
*Ewen. Bibliography of Eighteenth Century English Literature. Columbia University. 1935
*Marcuse. A Reference Guide for English Studies. University of Calfornia. 1990
*Bracken. Reference Works in British and American Literature. Libraries Unlimited. 1998
*Kennedy and Sands. A Concise Bibliography for Students of English. Stanford University
*Pelican Guide to English Literature
*[[w:The Cambridge History of English and American Literature|Cambridge History of English and American Literature]]
*[[s:The Cambridge History of English Literature|Cambridge History of English Literature]]
*Concise Cambridge History of English Literature
*New Cambridge History of English Literature
*Oxford History of English Literature
*Oxford Illustrated History of English Literature
*Short Oxford History of English Literature
*Routledge History of Literature in English
*Cambridge History of Early Medieval English Literature
*Cambridge History of Medieval English Literature
*Cambridge History of Early Modern English Literature
*Cambridge History of Victorian Literature
*Cambridge History of Twentieth Century English Literature
*[[s:The Cambridge History of American Literature|Cambridge History of American Literature]]

Periodicals
*Liverpool Magazine (1890)
*Literary Garner (1835)

===United States===
See [[w:Category:American literature by state]]
*Dershem. An Outline of American State Literature. 1921

Arizona:
*Joseph Amasa Munk, History of Arizona Literature, 1925
*Mary G Boyer, Arizona in Literature, 1935
*Etulain, &quot;Contours of Culture in Arizona and the Modern West&quot; in Luey and Stowe, Arizona at  Seventy-five, 1987
*Diaz, &quot;A Bibliography of Bibliographies Relating to the History and Literature of Arizona and New Mexico&quot; (1958) 14 Arizona Quarterly 197

Colorado:
*Eugene Parsons, &quot;Colorado Literature&quot; in Stone, History of Colorado, 1918, volume 1, chapter 42, p 877 [https://books.google.co.uk/books?id=-uVYAAAAMAAJ]
*Levette J Davidson, &quot;The Literature of Colorado&quot; in Hafen. Colorado and Its People: A Narrative and Topical History of the Centennial State. Volume 2. Chapter 8. Page 225 [https://books.google.co.uk/books?id=sncXAAAAIAAJ]
*Fritz. &quot;Literature&quot;. Colorado, the Centennial State. 1941. p 417 [https://books.google.co.uk/books?id=EU0UAAAAYAAJ]
*&quot;Books and Writers&quot;. Colorado, a Guide to the Highest State. 1941. p 96 [https://books.google.co.uk/books?id=3o8GAQAAIAAJ]
*Eugene Parsons, &quot;The Study of Colorado Literature&quot; (1918) Colorado School Journal, vols 34-35, p 24 [https://books.google.co.uk/books?id=QxNRAQAAMAAJ]
*&quot;A Plea for the Study of Colorado Literature&quot; (1918) The Trail: A Magazine &quot;for Colorado&quot;, vol 11, p 12 [https://books.google.co.uk/books?id=K7QTAAAAYAAJ]

Oregon:
*Powers, Alfred. History of Oregon Literature. Metropolitan Press. 1935 [https://books.google.co.uk/books?id=JzELAAAAMAAJ]
*Horner, John B. Oregon Literature. 1899 [https://books.google.co.uk/books?id=3ecWSpoFMcQC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==French==
See [[s:Category:French literature]]

Bibliographies and bibliographical works:
*A Critical Bibliography of French Literature. Syracuse University Press. [https://books.google.co.uk/books?id=IFJQl7eUrg4C&amp;pg=PR3#v=onepage&amp;q&amp;f=false vol 5]
*Bassan, Breed and Spinelli. An Annotated Bibliography of French Language and Literature. 2nd Ed: 1976 [https://books.google.co.uk/books?id=BdkaAAAAMAAJ]
*French XX Bibliography: A Bibliography for the Study of French Literature and Culture since 1885. [https://books.google.co.uk/books?id=VxVxFxyDOmkC&amp;pg=PA19125#v=onepage&amp;q&amp;f=false]
*Foulet. A Bibliography of Medieval French Literature for College Libraries. 1915 [https://books.google.co.uk/books?id=httNAQAAIAAJ]
*Kirsop. The Bibliography of French Literary History: Progress, Problems, Projects. 1964. [https://books.google.co.uk/books?id=Xt7nAAAAMAAJ]

*Jaffe. Bibliography of French Literature in American Magazines in the 18th Century. Michigan State College Press. 1951. [https://books.google.co.uk/books?id=Cy7GGsr1NI8C]
*Raimbert. French Literature in Mauritius (1800-1979): A Select Bibliography in the City Library of Port Louis. 1980 [https://books.google.co.uk/books?id=eQYpAQAAIAAJ]
*French Literature in Early American Translation: A Bibliographical Survey of Books and Pamphlets Printed in the United States from 1668 Through 1820. 1977. [https://books.google.co.uk/books?id=CYI0AQAAIAAJ]

History:
*Cambridge History of French Literature
*Coward. A History of French Literature: From Chanson de geste to Cinema. 2002. Paperback 2004. [https://books.google.co.uk/books?id=K8uS9vLpwuYC&amp;lpg=PP1&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Hollier. A New History of French Literature. 1989. 1994. [https://books.google.co.uk/books?id=nGQOodBVG9YC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Cazamian. A History of French Literature. Clarendon Press.1955. Reprinted 1967. [https://books.google.co.uk/books?id=W5PwAAAAMAAJ]
*Nitze and Dargan. A History of French Literature: From the Earliest Times to the Present. 1930. [https://books.google.co.uk/books?id=QT0nAAAAMAAJ]
*Butler. A History of French Literature. 1923. Reissued 1966. [https://books.google.co.uk/books?id=sPAoAAAAYAAJ]
*Dowden. A History of French Literature. (Literatures of the World). 1900 [https://books.google.co.uk/books?id=XWdcAAAAMAAJ]
*Wright. A History of French Literature. (Oxford French series). [https://books.google.co.uk/books?id=_O0oAAAAYAAJ]
*Schwarz. An Outline History of French Literature. 1924. 1932. [https://books.google.co.uk/books?id=9_fnAAAAMAAJ]
*Demogeot. History of French Literature. Adapted from the French by Bridge. 1874. [https://books.google.co.uk/books?id=LgxONYxlEogC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]

*Brunetière. Manual of the History of French Literature. 1898. [https://books.google.co.uk/books?id=WOHnAAAAMAAJ]

*Brereton. A Short History of French Literature. 1954. 2nd Ed:1976. [https://books.google.co.uk/books?id=DegoAAAAYAAJ]
*Bisson. A Short History of French Literature: From the Middle Ages to the Present Day. 1943. [https://books.google.co.uk/books?id=TLQywgEACAAJ]
*Hudson and Jack. A Short History of French Literature. 1919 [https://books.google.co.uk/books?id=8xIPAAAAQAAJ]
*Saintsbury. A Short History of French Literature. 1882. 2nd Ed: 1884. 7th Ed: 1917 [https://books.google.co.uk/books?id=WDoTAAAAMAAJ] [https://books.google.co.uk/books?id=AWFcAAAAMAAJ]

*Finch. French Literature: A Cultural History. 2010. [https://books.google.co.uk/books?id=8L0Z8uYUWj0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Prendergast. History of Modern French Literature: From the Sixteenth Century to the Twentieth Century. 2017. [https://books.google.co.uk/books?id=2QtpDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Laun. History of French Literature: From the Classical Renaissance until the End of the Reign of Louis XIV. 1883. [https://books.google.co.uk/books?id=bdZEAQAAIAAJ]

*Farrant. Introduction to Nineteenth-Century French Literature. 2007. [https://books.google.co.uk/books?id=m4HjBAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Vinet. History of French Literature in the Eighteenth Century. Translated by Bryce. 1854. [https://books.google.co.uk/books?id=5M803wEuBswC&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Jacqueline Cerquiglini-Toulet. A New History of Medieval French Literature. Translated by Sara Preisig. 2011. [https://books.google.co.uk/books?id=h42MQw6TfAcC&amp;pg=PT3#v=onepage&amp;q&amp;f=false]

*Konta. The History of French Literature: From the Oath of Strasburg to Chanticler. 1914. [https://books.google.co.uk/books?id=g2FcAAAAMAAJ]

*Cambridge Companion to Medieval French Literature

*Holmes. A History of Old French Literature: From the origins to 1300. 1938. Revised Ed: 1962. [https://books.google.co.uk/books?id=0UxcAAAAMAAJ]

*Sainstbury. A History of the French Novel (to the Close of the 19th Century). 1919. [https://books.google.co.uk/books?id=GAxJAQAAIAAJ]

==Japanese==
*Ryōzō Matsumoto. Japanese Literature New and Old. Hokuseido Press. 1966. [https://books.google.co.uk/books?id=EzftWTtGXzgC]
*J Ingram Bryan. The Literature of Japan. 1929. Kennikat Press. Port Washington. Reissued 1970. [https://books.google.co.uk/books?id=BwazAAAAIAAJ]
*Clay MacCauley. Japanese Literature. 1898. [https://books.google.co.uk/books?id=9nYuAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Epiphanius Wilson. Japanese Literature: Including Selections from Genji Monogatari and Classical Poetry and Drama of Japan. 1900. [https://books.google.co.uk/books?id=Iy4NAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Katsuhiko Takeda. Essays on Japanese Literature. Waseda University Press. 1977. [https://books.google.co.uk/books?id=VKkPAAAAYAAJ]

Bibliography
*Modern Japanese Literature in Western Translations: A Bibliography. International House of Japan Library. [https://books.google.co.uk/books?id=WZ7QAAAAMAAJ]

Periodicals
*Japanese Literature Today [https://books.google.co.uk/books?id=NyYHAQAAIAAJ]

Kokubungaku and nihonbungaku
*Tomoko Aoyama. &quot;From national literature to multicultural literature in Japanese language&quot;. Kaori Okano and Yoshio Sugimoto (eds). Rethinking Japanese Studies: Eurocentrism and the Asia-Pacific Region. Routledge Contemporary Japan Series. Chapter 4. pp [https://books.google.co.uk/books?id=sEcrDwAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false 53] to 72.

Reviewed
*Donald Richie. Japanese Literature Reviewed. 2003. [https://books.google.co.uk/books?id=ejJmAAAAMAAJ]

History
*Shuichi Kato. A History of Japanese Literature: From the Man'yōshū to Modern Times. New Abridged Edition. Japan Library. 1997. [https://books.google.co.uk/books?id=wUxOuD0NS5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Edward Putzar. Japanese Literature: A Historical Outline. University of Arizona Press. [https://books.google.co.uk/books?id=2kiBAAAAIAAJ]
*W G Aston. A History of Japanese Literature. (Literatures of the World). 1903. [https://books.google.co.uk/books?id=T4EMAAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Contemporary
*Kokusai Bunka Shinkokai (Japan Cultural Society). Introduction to Contemporary Japanese Literature: Synopese of Major Works: 1956-1970. University of Tokyo Press. 1972. [https://books.google.co.uk/books?id=XPQkHx6wR4IC]

Modern
*Routledge Handbook of Modern Japanese Literature. 2016. [https://books.google.co.uk/books?id=EMpJDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*J Thomas Rimer and Van C Gessel (eds). The Columbia Anthology of Modern Japanese Literature: Abridged. 2011. [https://books.google.co.uk/books?id=VrEYtVFv67oC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1].
*Karatani Kōjin. Origins of Modern Japanese Literature. 1993. [https://books.google.co.uk/books?id=hPJO2vEQgjYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Kan Kikuchi. History and Trends of Modern Japanese Literature. Tokyo. 1936. [https://books.google.co.uk/books?id=TU4DAAAAMAAJ]

Meiji and Taisho
*Kimura Ki (editor and compiler). Japanese Literature: Manners and Customs in the Meiji-Taishó Era. Ōbunsha. [https://books.google.co.uk/books?id=rBQrAAAAIAAJ]

Early modern
*Haruo Shirane (ed). Early Modern Japanese Literature: An Anthology, 1600-1900. Abridged Edition. 2008. [https://books.google.co.uk/books?id=SN72QCVBpVAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Classical
*The Princeton Companion to Classical Japanese Literature [https://books.google.co.uk/books?id=BSmMbQhafJoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Introduction to Classic Japanese Literature. Kokusai Bunka Shinkokai. 1948. [https://books.google.co.uk/books?id=WJAPAAAAYAAJ]

Traditional
*Haruo Shirane (ed). Traditional Japanese Literature: An Anthology, Beginnings to 1600. 2007. [https://books.google.co.uk/books?id=LsHfIsIXgEgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Literary criticism; Literary studies
*Seth Jacobowitz and Jonathan E Abel (eds). Modern Japanese Literary Studies. 2026. [https://books.google.co.uk/books?id=s-_AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Studies in Modern Japanese Literature: Essays and Translations in Honor of Edwin McClellan. 1997. [https://books.google.co.uk/books?id=_5EPAAAAYAAJ]

Anthology
*Donald Keene. Anthology of Japanese Literature from the earliest era to the mid-nineteenth century. Grove Press. 1955. [https://books.google.co.uk/books?id=9h8tAAAAMAAJ] Great Britain. 1956. [https://books.google.co.uk/books?id=LCxkAAAAMAAJ]. Evergreen Ed: 1960. [https://books.google.co.uk/books?id=rumBAAAAIAAJ]

[[Category:Literature]]</text>
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      <text bytes="13578" sha1="4bxmbogx2xyjbgs61j53js9wd5sho2n" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Bibliography|Bibliography]]

This part of the [[Universal Bibliography]] is a bibliography of literature.

See [[s:Category:Literature]], [[s:Category:History of literature]], [[w:Bibliography of encyclopedias: literature]] and [[w:Category:Works about literature]]

==World==
*Cassell's Encyclopaedia of World Literature. 1953. 2nd Ed: 1973. vol 1 (histories and general articles): [https://books.google.co.uk/books?id=soIYAAAAIAAJ]. vol 3: [https://books.google.co.uk/books?id=AIEYAAAAIAAJ]. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA197#v=onepage&amp;q&amp;f=false].

*Damrosch. What is World Literature? 2003. [https://books.google.co.uk/books?id=yY-17mtp9R8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series:
*Edmund Gosse (ed). Literatures of the World. D Appleton and Co [https://en.m.wikisource.org/wiki/Page:A_history_of_Bohemian_literature.pdf/444]

==English==
*Bateson. Cambridge Bibliography of English Literature. 1940. Watson, Supplement 1957.
*Concise Cambridge Bibliography of English Literature
*New Cambridge Bibliography of English Literature
*Annual Bibliography of English Language and Literature. Cambridge University
*Ewen. Bibliography of Eighteenth Century English Literature. Columbia University. 1935
*Marcuse. A Reference Guide for English Studies. University of Calfornia. 1990
*Bracken. Reference Works in British and American Literature. Libraries Unlimited. 1998
*Kennedy and Sands. A Concise Bibliography for Students of English. Stanford University
*Pelican Guide to English Literature
*[[w:The Cambridge History of English and American Literature|Cambridge History of English and American Literature]]
*[[s:The Cambridge History of English Literature|Cambridge History of English Literature]]
*Concise Cambridge History of English Literature
*New Cambridge History of English Literature
*Oxford History of English Literature
*Oxford Illustrated History of English Literature
*Short Oxford History of English Literature
*Routledge History of Literature in English
*Cambridge History of Early Medieval English Literature
*Cambridge History of Medieval English Literature
*Cambridge History of Early Modern English Literature
*Cambridge History of Victorian Literature
*Cambridge History of Twentieth Century English Literature
*[[s:The Cambridge History of American Literature|Cambridge History of American Literature]]

Periodicals
*Liverpool Magazine (1890)
*Literary Garner (1835)

===United States===
See [[w:Category:American literature by state]]
*Dershem. An Outline of American State Literature. 1921

Arizona:
*Joseph Amasa Munk, History of Arizona Literature, 1925
*Mary G Boyer, Arizona in Literature, 1935
*Etulain, &quot;Contours of Culture in Arizona and the Modern West&quot; in Luey and Stowe, Arizona at  Seventy-five, 1987
*Diaz, &quot;A Bibliography of Bibliographies Relating to the History and Literature of Arizona and New Mexico&quot; (1958) 14 Arizona Quarterly 197

Colorado:
*Eugene Parsons, &quot;Colorado Literature&quot; in Stone, History of Colorado, 1918, volume 1, chapter 42, p 877 [https://books.google.co.uk/books?id=-uVYAAAAMAAJ]
*Levette J Davidson, &quot;The Literature of Colorado&quot; in Hafen. Colorado and Its People: A Narrative and Topical History of the Centennial State. Volume 2. Chapter 8. Page 225 [https://books.google.co.uk/books?id=sncXAAAAIAAJ]
*Fritz. &quot;Literature&quot;. Colorado, the Centennial State. 1941. p 417 [https://books.google.co.uk/books?id=EU0UAAAAYAAJ]
*&quot;Books and Writers&quot;. Colorado, a Guide to the Highest State. 1941. p 96 [https://books.google.co.uk/books?id=3o8GAQAAIAAJ]
*Eugene Parsons, &quot;The Study of Colorado Literature&quot; (1918) Colorado School Journal, vols 34-35, p 24 [https://books.google.co.uk/books?id=QxNRAQAAMAAJ]
*&quot;A Plea for the Study of Colorado Literature&quot; (1918) The Trail: A Magazine &quot;for Colorado&quot;, vol 11, p 12 [https://books.google.co.uk/books?id=K7QTAAAAYAAJ]

Oregon:
*Powers, Alfred. History of Oregon Literature. Metropolitan Press. 1935 [https://books.google.co.uk/books?id=JzELAAAAMAAJ]
*Horner, John B. Oregon Literature. 1899 [https://books.google.co.uk/books?id=3ecWSpoFMcQC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==French==
See [[s:Category:French literature]]

Bibliographies and bibliographical works:
*A Critical Bibliography of French Literature. Syracuse University Press. [https://books.google.co.uk/books?id=IFJQl7eUrg4C&amp;pg=PR3#v=onepage&amp;q&amp;f=false vol 5]
*Bassan, Breed and Spinelli. An Annotated Bibliography of French Language and Literature. 2nd Ed: 1976 [https://books.google.co.uk/books?id=BdkaAAAAMAAJ]
*French XX Bibliography: A Bibliography for the Study of French Literature and Culture since 1885. [https://books.google.co.uk/books?id=VxVxFxyDOmkC&amp;pg=PA19125#v=onepage&amp;q&amp;f=false]
*Foulet. A Bibliography of Medieval French Literature for College Libraries. 1915 [https://books.google.co.uk/books?id=httNAQAAIAAJ]
*Kirsop. The Bibliography of French Literary History: Progress, Problems, Projects. 1964. [https://books.google.co.uk/books?id=Xt7nAAAAMAAJ]

*Jaffe. Bibliography of French Literature in American Magazines in the 18th Century. Michigan State College Press. 1951. [https://books.google.co.uk/books?id=Cy7GGsr1NI8C]
*Raimbert. French Literature in Mauritius (1800-1979): A Select Bibliography in the City Library of Port Louis. 1980 [https://books.google.co.uk/books?id=eQYpAQAAIAAJ]
*French Literature in Early American Translation: A Bibliographical Survey of Books and Pamphlets Printed in the United States from 1668 Through 1820. 1977. [https://books.google.co.uk/books?id=CYI0AQAAIAAJ]

History:
*Cambridge History of French Literature
*Coward. A History of French Literature: From Chanson de geste to Cinema. 2002. Paperback 2004. [https://books.google.co.uk/books?id=K8uS9vLpwuYC&amp;lpg=PP1&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Hollier. A New History of French Literature. 1989. 1994. [https://books.google.co.uk/books?id=nGQOodBVG9YC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Cazamian. A History of French Literature. Clarendon Press.1955. Reprinted 1967. [https://books.google.co.uk/books?id=W5PwAAAAMAAJ]
*Nitze and Dargan. A History of French Literature: From the Earliest Times to the Present. 1930. [https://books.google.co.uk/books?id=QT0nAAAAMAAJ]
*Butler. A History of French Literature. 1923. Reissued 1966. [https://books.google.co.uk/books?id=sPAoAAAAYAAJ]
*Dowden. A History of French Literature. (Literatures of the World). 1900 [https://books.google.co.uk/books?id=XWdcAAAAMAAJ]
*Wright. A History of French Literature. (Oxford French series). [https://books.google.co.uk/books?id=_O0oAAAAYAAJ]
*Schwarz. An Outline History of French Literature. 1924. 1932. [https://books.google.co.uk/books?id=9_fnAAAAMAAJ]
*Demogeot. History of French Literature. Adapted from the French by Bridge. 1874. [https://books.google.co.uk/books?id=LgxONYxlEogC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]

*Brunetière. Manual of the History of French Literature. 1898. [https://books.google.co.uk/books?id=WOHnAAAAMAAJ]

*Brereton. A Short History of French Literature. 1954. 2nd Ed:1976. [https://books.google.co.uk/books?id=DegoAAAAYAAJ]
*Bisson. A Short History of French Literature: From the Middle Ages to the Present Day. 1943. [https://books.google.co.uk/books?id=TLQywgEACAAJ]
*Hudson and Jack. A Short History of French Literature. 1919 [https://books.google.co.uk/books?id=8xIPAAAAQAAJ]
*Saintsbury. A Short History of French Literature. 1882. 2nd Ed: 1884. 7th Ed: 1917 [https://books.google.co.uk/books?id=WDoTAAAAMAAJ] [https://books.google.co.uk/books?id=AWFcAAAAMAAJ]

*Finch. French Literature: A Cultural History. 2010. [https://books.google.co.uk/books?id=8L0Z8uYUWj0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Prendergast. History of Modern French Literature: From the Sixteenth Century to the Twentieth Century. 2017. [https://books.google.co.uk/books?id=2QtpDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Laun. History of French Literature: From the Classical Renaissance until the End of the Reign of Louis XIV. 1883. [https://books.google.co.uk/books?id=bdZEAQAAIAAJ]

*Farrant. Introduction to Nineteenth-Century French Literature. 2007. [https://books.google.co.uk/books?id=m4HjBAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Vinet. History of French Literature in the Eighteenth Century. Translated by Bryce. 1854. [https://books.google.co.uk/books?id=5M803wEuBswC&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Jacqueline Cerquiglini-Toulet. A New History of Medieval French Literature. Translated by Sara Preisig. 2011. [https://books.google.co.uk/books?id=h42MQw6TfAcC&amp;pg=PT3#v=onepage&amp;q&amp;f=false]

*Konta. The History of French Literature: From the Oath of Strasburg to Chanticler. 1914. [https://books.google.co.uk/books?id=g2FcAAAAMAAJ]

*Cambridge Companion to Medieval French Literature

*Holmes. A History of Old French Literature: From the origins to 1300. 1938. Revised Ed: 1962. [https://books.google.co.uk/books?id=0UxcAAAAMAAJ]

*Sainstbury. A History of the French Novel (to the Close of the 19th Century). 1919. [https://books.google.co.uk/books?id=GAxJAQAAIAAJ]

==Japanese==
*Ryōzō Matsumoto. Japanese Literature New and Old. Hokuseido Press. 1966. [https://books.google.co.uk/books?id=EzftWTtGXzgC]
*J Ingram Bryan. The Literature of Japan. 1929. Kennikat Press. Port Washington. Reissued 1970. [https://books.google.co.uk/books?id=BwazAAAAIAAJ]
*Clay MacCauley. Japanese Literature. 1898. [https://books.google.co.uk/books?id=9nYuAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Epiphanius Wilson. Japanese Literature: Including Selections from Genji Monogatari and Classical Poetry and Drama of Japan. 1900. [https://books.google.co.uk/books?id=Iy4NAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Katsuhiko Takeda. Essays on Japanese Literature. Waseda University Press. 1977. [https://books.google.co.uk/books?id=VKkPAAAAYAAJ]

Bibliography
*Modern Japanese Literature in Western Translations: A Bibliography. International House of Japan Library. [https://books.google.co.uk/books?id=WZ7QAAAAMAAJ]

Periodicals
*Japanese Literature Today [https://books.google.co.uk/books?id=NyYHAQAAIAAJ]

Kokubungaku and nihonbungaku
*Tomoko Aoyama. &quot;From national literature to multicultural literature in Japanese language&quot;. Kaori Okano and Yoshio Sugimoto (eds). Rethinking Japanese Studies: Eurocentrism and the Asia-Pacific Region. Routledge Contemporary Japan Series. Chapter 4. pp [https://books.google.co.uk/books?id=sEcrDwAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false 53] to 72.

Reviewed
*Donald Richie. Japanese Literature Reviewed. 2003. [https://books.google.co.uk/books?id=ejJmAAAAMAAJ]

History
*Shuichi Kato. A History of Japanese Literature: From the Man'yōshū to Modern Times. New Abridged Edition. Japan Library. 1997. [https://books.google.co.uk/books?id=wUxOuD0NS5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Edward Putzar. Japanese Literature: A Historical Outline. University of Arizona Press. [https://books.google.co.uk/books?id=2kiBAAAAIAAJ]
*W G Aston. A History of Japanese Literature. (Literatures of the World). 1903. [https://books.google.co.uk/books?id=T4EMAAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Naomi Fukuda (ed). Literature. (Japanese History: A Guide to Survey Histories, Part 2). Center for Japanese Studies, University of Michigan. 1984. [https://books.google.co.uk/books?id=QsoUAQAAIAAJ]

Contemporary
*Kokusai Bunka Shinkokai (Japan Cultural Society). Introduction to Contemporary Japanese Literature: Synopese of Major Works: 1956-1970. University of Tokyo Press. 1972. [https://books.google.co.uk/books?id=XPQkHx6wR4IC]

Modern
*Routledge Handbook of Modern Japanese Literature. 2016. [https://books.google.co.uk/books?id=EMpJDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*J Thomas Rimer and Van C Gessel (eds). The Columbia Anthology of Modern Japanese Literature: Abridged. 2011. [https://books.google.co.uk/books?id=VrEYtVFv67oC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1].
*Karatani Kōjin. Origins of Modern Japanese Literature. 1993. [https://books.google.co.uk/books?id=hPJO2vEQgjYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Kan Kikuchi. History and Trends of Modern Japanese Literature. Tokyo. 1936. [https://books.google.co.uk/books?id=TU4DAAAAMAAJ]

Meiji and Taisho
*Kimura Ki (editor and compiler). Japanese Literature: Manners and Customs in the Meiji-Taishó Era. Ōbunsha. [https://books.google.co.uk/books?id=rBQrAAAAIAAJ]

Early modern
*Haruo Shirane (ed). Early Modern Japanese Literature: An Anthology, 1600-1900. Abridged Edition. 2008. [https://books.google.co.uk/books?id=SN72QCVBpVAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Classical
*The Princeton Companion to Classical Japanese Literature [https://books.google.co.uk/books?id=BSmMbQhafJoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Introduction to Classic Japanese Literature. Kokusai Bunka Shinkokai. 1948. [https://books.google.co.uk/books?id=WJAPAAAAYAAJ]

Traditional
*Haruo Shirane (ed). Traditional Japanese Literature: An Anthology, Beginnings to 1600. 2007. [https://books.google.co.uk/books?id=LsHfIsIXgEgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Literary criticism; Literary studies
*Seth Jacobowitz and Jonathan E Abel (eds). Modern Japanese Literary Studies. 2026. [https://books.google.co.uk/books?id=s-_AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Studies in Modern Japanese Literature: Essays and Translations in Honor of Edwin McClellan. 1997. [https://books.google.co.uk/books?id=_5EPAAAAYAAJ]

Anthology
*Donald Keene. Anthology of Japanese Literature from the earliest era to the mid-nineteenth century. Grove Press. 1955. [https://books.google.co.uk/books?id=9h8tAAAAMAAJ] Great Britain. 1956. [https://books.google.co.uk/books?id=LCxkAAAAMAAJ]. Evergreen Ed: 1960. [https://books.google.co.uk/books?id=rumBAAAAIAAJ]

[[Category:Literature]]</text>
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      <text bytes="13671" sha1="n52yw3d0tjfjasjsbpe8oyvleo89pp6" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Bibliography|Bibliography]]

This part of the [[Universal Bibliography]] is a bibliography of literature.

See [[s:Category:Literature]], [[s:Category:History of literature]], [[w:Bibliography of encyclopedias: literature]] and [[w:Category:Works about literature]]

==World==
*Cassell's Encyclopaedia of World Literature. 1953. 2nd Ed: 1973. vol 1 (histories and general articles): [https://books.google.co.uk/books?id=soIYAAAAIAAJ]. vol 3: [https://books.google.co.uk/books?id=AIEYAAAAIAAJ]. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA197#v=onepage&amp;q&amp;f=false].

*Damrosch. What is World Literature? 2003. [https://books.google.co.uk/books?id=yY-17mtp9R8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series:
*Edmund Gosse (ed). Literatures of the World. D Appleton and Co [https://en.m.wikisource.org/wiki/Page:A_history_of_Bohemian_literature.pdf/444]

==English==
*Bateson. Cambridge Bibliography of English Literature. 1940. Watson, Supplement 1957.
*Concise Cambridge Bibliography of English Literature
*New Cambridge Bibliography of English Literature
*Annual Bibliography of English Language and Literature. Cambridge University
*Ewen. Bibliography of Eighteenth Century English Literature. Columbia University. 1935
*Marcuse. A Reference Guide for English Studies. University of Calfornia. 1990
*Bracken. Reference Works in British and American Literature. Libraries Unlimited. 1998
*Kennedy and Sands. A Concise Bibliography for Students of English. Stanford University
*Pelican Guide to English Literature
*[[w:The Cambridge History of English and American Literature|Cambridge History of English and American Literature]]
*[[s:The Cambridge History of English Literature|Cambridge History of English Literature]]
*Concise Cambridge History of English Literature
*New Cambridge History of English Literature
*Oxford History of English Literature
*Oxford Illustrated History of English Literature
*Short Oxford History of English Literature
*Routledge History of Literature in English
*Cambridge History of Early Medieval English Literature
*Cambridge History of Medieval English Literature
*Cambridge History of Early Modern English Literature
*Cambridge History of Victorian Literature
*Cambridge History of Twentieth Century English Literature
*[[s:The Cambridge History of American Literature|Cambridge History of American Literature]]

Periodicals
*Liverpool Magazine (1890)
*Literary Garner (1835)

===United States===
See [[w:Category:American literature by state]]
*Dershem. An Outline of American State Literature. 1921

Arizona:
*Joseph Amasa Munk, History of Arizona Literature, 1925
*Mary G Boyer, Arizona in Literature, 1935
*Etulain, &quot;Contours of Culture in Arizona and the Modern West&quot; in Luey and Stowe, Arizona at  Seventy-five, 1987
*Diaz, &quot;A Bibliography of Bibliographies Relating to the History and Literature of Arizona and New Mexico&quot; (1958) 14 Arizona Quarterly 197

Colorado:
*Eugene Parsons, &quot;Colorado Literature&quot; in Stone, History of Colorado, 1918, volume 1, chapter 42, p 877 [https://books.google.co.uk/books?id=-uVYAAAAMAAJ]
*Levette J Davidson, &quot;The Literature of Colorado&quot; in Hafen. Colorado and Its People: A Narrative and Topical History of the Centennial State. Volume 2. Chapter 8. Page 225 [https://books.google.co.uk/books?id=sncXAAAAIAAJ]
*Fritz. &quot;Literature&quot;. Colorado, the Centennial State. 1941. p 417 [https://books.google.co.uk/books?id=EU0UAAAAYAAJ]
*&quot;Books and Writers&quot;. Colorado, a Guide to the Highest State. 1941. p 96 [https://books.google.co.uk/books?id=3o8GAQAAIAAJ]
*Eugene Parsons, &quot;The Study of Colorado Literature&quot; (1918) Colorado School Journal, vols 34-35, p 24 [https://books.google.co.uk/books?id=QxNRAQAAMAAJ]
*&quot;A Plea for the Study of Colorado Literature&quot; (1918) The Trail: A Magazine &quot;for Colorado&quot;, vol 11, p 12 [https://books.google.co.uk/books?id=K7QTAAAAYAAJ]

Oregon:
*Powers, Alfred. History of Oregon Literature. Metropolitan Press. 1935 [https://books.google.co.uk/books?id=JzELAAAAMAAJ]
*Horner, John B. Oregon Literature. 1899 [https://books.google.co.uk/books?id=3ecWSpoFMcQC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==French==
See [[s:Category:French literature]]

Bibliographies and bibliographical works:
*A Critical Bibliography of French Literature. Syracuse University Press. [https://books.google.co.uk/books?id=IFJQl7eUrg4C&amp;pg=PR3#v=onepage&amp;q&amp;f=false vol 5]
*Bassan, Breed and Spinelli. An Annotated Bibliography of French Language and Literature. 2nd Ed: 1976 [https://books.google.co.uk/books?id=BdkaAAAAMAAJ]
*French XX Bibliography: A Bibliography for the Study of French Literature and Culture since 1885. [https://books.google.co.uk/books?id=VxVxFxyDOmkC&amp;pg=PA19125#v=onepage&amp;q&amp;f=false]
*Foulet. A Bibliography of Medieval French Literature for College Libraries. 1915 [https://books.google.co.uk/books?id=httNAQAAIAAJ]
*Kirsop. The Bibliography of French Literary History: Progress, Problems, Projects. 1964. [https://books.google.co.uk/books?id=Xt7nAAAAMAAJ]

*Jaffe. Bibliography of French Literature in American Magazines in the 18th Century. Michigan State College Press. 1951. [https://books.google.co.uk/books?id=Cy7GGsr1NI8C]
*Raimbert. French Literature in Mauritius (1800-1979): A Select Bibliography in the City Library of Port Louis. 1980 [https://books.google.co.uk/books?id=eQYpAQAAIAAJ]
*French Literature in Early American Translation: A Bibliographical Survey of Books and Pamphlets Printed in the United States from 1668 Through 1820. 1977. [https://books.google.co.uk/books?id=CYI0AQAAIAAJ]

History:
*Cambridge History of French Literature
*Coward. A History of French Literature: From Chanson de geste to Cinema. 2002. Paperback 2004. [https://books.google.co.uk/books?id=K8uS9vLpwuYC&amp;lpg=PP1&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Hollier. A New History of French Literature. 1989. 1994. [https://books.google.co.uk/books?id=nGQOodBVG9YC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Cazamian. A History of French Literature. Clarendon Press.1955. Reprinted 1967. [https://books.google.co.uk/books?id=W5PwAAAAMAAJ]
*Nitze and Dargan. A History of French Literature: From the Earliest Times to the Present. 1930. [https://books.google.co.uk/books?id=QT0nAAAAMAAJ]
*Butler. A History of French Literature. 1923. Reissued 1966. [https://books.google.co.uk/books?id=sPAoAAAAYAAJ]
*Dowden. A History of French Literature. (Literatures of the World). 1900 [https://books.google.co.uk/books?id=XWdcAAAAMAAJ]
*Wright. A History of French Literature. (Oxford French series). [https://books.google.co.uk/books?id=_O0oAAAAYAAJ]
*Schwarz. An Outline History of French Literature. 1924. 1932. [https://books.google.co.uk/books?id=9_fnAAAAMAAJ]
*Demogeot. History of French Literature. Adapted from the French by Bridge. 1874. [https://books.google.co.uk/books?id=LgxONYxlEogC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]

*Brunetière. Manual of the History of French Literature. 1898. [https://books.google.co.uk/books?id=WOHnAAAAMAAJ]

*Brereton. A Short History of French Literature. 1954. 2nd Ed:1976. [https://books.google.co.uk/books?id=DegoAAAAYAAJ]
*Bisson. A Short History of French Literature: From the Middle Ages to the Present Day. 1943. [https://books.google.co.uk/books?id=TLQywgEACAAJ]
*Hudson and Jack. A Short History of French Literature. 1919 [https://books.google.co.uk/books?id=8xIPAAAAQAAJ]
*Saintsbury. A Short History of French Literature. 1882. 2nd Ed: 1884. 7th Ed: 1917 [https://books.google.co.uk/books?id=WDoTAAAAMAAJ] [https://books.google.co.uk/books?id=AWFcAAAAMAAJ]

*Finch. French Literature: A Cultural History. 2010. [https://books.google.co.uk/books?id=8L0Z8uYUWj0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Prendergast. History of Modern French Literature: From the Sixteenth Century to the Twentieth Century. 2017. [https://books.google.co.uk/books?id=2QtpDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Laun. History of French Literature: From the Classical Renaissance until the End of the Reign of Louis XIV. 1883. [https://books.google.co.uk/books?id=bdZEAQAAIAAJ]

*Farrant. Introduction to Nineteenth-Century French Literature. 2007. [https://books.google.co.uk/books?id=m4HjBAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Vinet. History of French Literature in the Eighteenth Century. Translated by Bryce. 1854. [https://books.google.co.uk/books?id=5M803wEuBswC&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Jacqueline Cerquiglini-Toulet. A New History of Medieval French Literature. Translated by Sara Preisig. 2011. [https://books.google.co.uk/books?id=h42MQw6TfAcC&amp;pg=PT3#v=onepage&amp;q&amp;f=false]

*Konta. The History of French Literature: From the Oath of Strasburg to Chanticler. 1914. [https://books.google.co.uk/books?id=g2FcAAAAMAAJ]

*Cambridge Companion to Medieval French Literature

*Holmes. A History of Old French Literature: From the origins to 1300. 1938. Revised Ed: 1962. [https://books.google.co.uk/books?id=0UxcAAAAMAAJ]

*Sainstbury. A History of the French Novel (to the Close of the 19th Century). 1919. [https://books.google.co.uk/books?id=GAxJAQAAIAAJ]

==Japanese==
*Ryōzō Matsumoto. Japanese Literature New and Old. Hokuseido Press. 1966. [https://books.google.co.uk/books?id=EzftWTtGXzgC]
*J Ingram Bryan. The Literature of Japan. 1929. Kennikat Press. Port Washington. Reissued 1970. [https://books.google.co.uk/books?id=BwazAAAAIAAJ]
*Clay MacCauley. Japanese Literature. 1898. [https://books.google.co.uk/books?id=9nYuAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Epiphanius Wilson. Japanese Literature: Including Selections from Genji Monogatari and Classical Poetry and Drama of Japan. 1900. [https://books.google.co.uk/books?id=Iy4NAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Katsuhiko Takeda. Essays on Japanese Literature. Waseda University Press. 1977. [https://books.google.co.uk/books?id=VKkPAAAAYAAJ]

Bibliography
*Modern Japanese Literature in Western Translations: A Bibliography. International House of Japan Library. [https://books.google.co.uk/books?id=WZ7QAAAAMAAJ]

Periodicals
*Japanese Literature Today [https://books.google.co.uk/books?id=NyYHAQAAIAAJ]

Kokubungaku and nihonbungaku
*Tomoko Aoyama. &quot;From national literature to multicultural literature in Japanese language&quot;. Kaori Okano and Yoshio Sugimoto (eds). Rethinking Japanese Studies: Eurocentrism and the Asia-Pacific Region. Routledge Contemporary Japan Series. Chapter 4. pp [https://books.google.co.uk/books?id=sEcrDwAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false 53] to 72.

Reviewed
*Donald Richie. Japanese Literature Reviewed. 2003. [https://books.google.co.uk/books?id=ejJmAAAAMAAJ]

History
*Shuichi Kato. A History of Japanese Literature: From the Man'yōshū to Modern Times. New Abridged Edition. Japan Library. 1997. [https://books.google.co.uk/books?id=wUxOuD0NS5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Edward Putzar. Japanese Literature: A Historical Outline. University of Arizona Press. [https://books.google.co.uk/books?id=2kiBAAAAIAAJ]
*W G Aston. A History of Japanese Literature. (Literatures of the World). 1903. [https://books.google.co.uk/books?id=T4EMAAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Naomi Fukuda (ed). Literature. (Japanese History: A Guide to Survey Histories, Part 2). Center for Japanese Studies, University of Michigan. 1984. [https://books.google.co.uk/books?id=QsoUAQAAIAAJ]

Contemporary
*Kokusai Bunka Shinkokai (Japan Cultural Society). Introduction to Contemporary Japanese Literature: Synopese of Major Works: 1956-1970. University of Tokyo Press. 1972. [https://books.google.co.uk/books?id=XPQkHx6wR4IC]

Modern
*Routledge Handbook of Modern Japanese Literature. 2016. [https://books.google.co.uk/books?id=EMpJDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*J Thomas Rimer and Van C Gessel (eds). The Columbia Anthology of Modern Japanese Literature. 2007. vol 2. [https://books.google.co.uk/books?id=BAg9tUJR1aEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
**Abridged. 2011. [https://books.google.co.uk/books?id=VrEYtVFv67oC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1].
*Karatani Kōjin. Origins of Modern Japanese Literature. 1993. [https://books.google.co.uk/books?id=hPJO2vEQgjYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Kan Kikuchi. History and Trends of Modern Japanese Literature. Tokyo. 1936. [https://books.google.co.uk/books?id=TU4DAAAAMAAJ]

Meiji and Taisho
*Kimura Ki (editor and compiler). Japanese Literature: Manners and Customs in the Meiji-Taishó Era. Ōbunsha. [https://books.google.co.uk/books?id=rBQrAAAAIAAJ]

Early modern
*Haruo Shirane (ed). Early Modern Japanese Literature: An Anthology, 1600-1900. Abridged Edition. 2008. [https://books.google.co.uk/books?id=SN72QCVBpVAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Classical
*The Princeton Companion to Classical Japanese Literature [https://books.google.co.uk/books?id=BSmMbQhafJoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Introduction to Classic Japanese Literature. Kokusai Bunka Shinkokai. 1948. [https://books.google.co.uk/books?id=WJAPAAAAYAAJ]

Traditional
*Haruo Shirane (ed). Traditional Japanese Literature: An Anthology, Beginnings to 1600. 2007. [https://books.google.co.uk/books?id=LsHfIsIXgEgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Literary criticism; Literary studies
*Seth Jacobowitz and Jonathan E Abel (eds). Modern Japanese Literary Studies. 2026. [https://books.google.co.uk/books?id=s-_AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Studies in Modern Japanese Literature: Essays and Translations in Honor of Edwin McClellan. 1997. [https://books.google.co.uk/books?id=_5EPAAAAYAAJ]

Anthology
*Donald Keene. Anthology of Japanese Literature from the earliest era to the mid-nineteenth century. Grove Press. 1955. [https://books.google.co.uk/books?id=9h8tAAAAMAAJ] Great Britain. 1956. [https://books.google.co.uk/books?id=LCxkAAAAMAAJ]. Evergreen Ed: 1960. [https://books.google.co.uk/books?id=rumBAAAAIAAJ]

[[Category:Literature]]</text>
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      <text bytes="13840" sha1="jgmdzd4rcw7wmfjq315hehlc9ukf34k" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Bibliography|Bibliography]]

This part of the [[Universal Bibliography]] is a bibliography of literature.

See [[s:Category:Literature]], [[s:Category:History of literature]], [[w:Bibliography of encyclopedias: literature]] and [[w:Category:Works about literature]]

==World==
*Cassell's Encyclopaedia of World Literature. 1953. 2nd Ed: 1973. vol 1 (histories and general articles): [https://books.google.co.uk/books?id=soIYAAAAIAAJ]. vol 3: [https://books.google.co.uk/books?id=AIEYAAAAIAAJ]. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA197#v=onepage&amp;q&amp;f=false].

*Damrosch. What is World Literature? 2003. [https://books.google.co.uk/books?id=yY-17mtp9R8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series:
*Edmund Gosse (ed). Literatures of the World. D Appleton and Co [https://en.m.wikisource.org/wiki/Page:A_history_of_Bohemian_literature.pdf/444]

==English==
*Bateson. Cambridge Bibliography of English Literature. 1940. Watson, Supplement 1957.
*Concise Cambridge Bibliography of English Literature
*New Cambridge Bibliography of English Literature
*Annual Bibliography of English Language and Literature. Cambridge University
*Ewen. Bibliography of Eighteenth Century English Literature. Columbia University. 1935
*Marcuse. A Reference Guide for English Studies. University of Calfornia. 1990
*Bracken. Reference Works in British and American Literature. Libraries Unlimited. 1998
*Kennedy and Sands. A Concise Bibliography for Students of English. Stanford University
*Pelican Guide to English Literature
*[[w:The Cambridge History of English and American Literature|Cambridge History of English and American Literature]]
*[[s:The Cambridge History of English Literature|Cambridge History of English Literature]]
*Concise Cambridge History of English Literature
*New Cambridge History of English Literature
*Oxford History of English Literature
*Oxford Illustrated History of English Literature
*Short Oxford History of English Literature
*Routledge History of Literature in English
*Cambridge History of Early Medieval English Literature
*Cambridge History of Medieval English Literature
*Cambridge History of Early Modern English Literature
*Cambridge History of Victorian Literature
*Cambridge History of Twentieth Century English Literature
*[[s:The Cambridge History of American Literature|Cambridge History of American Literature]]

Periodicals
*Liverpool Magazine (1890)
*Literary Garner (1835)

===United States===
See [[w:Category:American literature by state]]
*Dershem. An Outline of American State Literature. 1921

Arizona:
*Joseph Amasa Munk, History of Arizona Literature, 1925
*Mary G Boyer, Arizona in Literature, 1935
*Etulain, &quot;Contours of Culture in Arizona and the Modern West&quot; in Luey and Stowe, Arizona at  Seventy-five, 1987
*Diaz, &quot;A Bibliography of Bibliographies Relating to the History and Literature of Arizona and New Mexico&quot; (1958) 14 Arizona Quarterly 197

Colorado:
*Eugene Parsons, &quot;Colorado Literature&quot; in Stone, History of Colorado, 1918, volume 1, chapter 42, p 877 [https://books.google.co.uk/books?id=-uVYAAAAMAAJ]
*Levette J Davidson, &quot;The Literature of Colorado&quot; in Hafen. Colorado and Its People: A Narrative and Topical History of the Centennial State. Volume 2. Chapter 8. Page 225 [https://books.google.co.uk/books?id=sncXAAAAIAAJ]
*Fritz. &quot;Literature&quot;. Colorado, the Centennial State. 1941. p 417 [https://books.google.co.uk/books?id=EU0UAAAAYAAJ]
*&quot;Books and Writers&quot;. Colorado, a Guide to the Highest State. 1941. p 96 [https://books.google.co.uk/books?id=3o8GAQAAIAAJ]
*Eugene Parsons, &quot;The Study of Colorado Literature&quot; (1918) Colorado School Journal, vols 34-35, p 24 [https://books.google.co.uk/books?id=QxNRAQAAMAAJ]
*&quot;A Plea for the Study of Colorado Literature&quot; (1918) The Trail: A Magazine &quot;for Colorado&quot;, vol 11, p 12 [https://books.google.co.uk/books?id=K7QTAAAAYAAJ]

Oregon:
*Powers, Alfred. History of Oregon Literature. Metropolitan Press. 1935 [https://books.google.co.uk/books?id=JzELAAAAMAAJ]
*Horner, John B. Oregon Literature. 1899 [https://books.google.co.uk/books?id=3ecWSpoFMcQC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==French==
See [[s:Category:French literature]]

Bibliographies and bibliographical works:
*A Critical Bibliography of French Literature. Syracuse University Press. [https://books.google.co.uk/books?id=IFJQl7eUrg4C&amp;pg=PR3#v=onepage&amp;q&amp;f=false vol 5]
*Bassan, Breed and Spinelli. An Annotated Bibliography of French Language and Literature. 2nd Ed: 1976 [https://books.google.co.uk/books?id=BdkaAAAAMAAJ]
*French XX Bibliography: A Bibliography for the Study of French Literature and Culture since 1885. [https://books.google.co.uk/books?id=VxVxFxyDOmkC&amp;pg=PA19125#v=onepage&amp;q&amp;f=false]
*Foulet. A Bibliography of Medieval French Literature for College Libraries. 1915 [https://books.google.co.uk/books?id=httNAQAAIAAJ]
*Kirsop. The Bibliography of French Literary History: Progress, Problems, Projects. 1964. [https://books.google.co.uk/books?id=Xt7nAAAAMAAJ]

*Jaffe. Bibliography of French Literature in American Magazines in the 18th Century. Michigan State College Press. 1951. [https://books.google.co.uk/books?id=Cy7GGsr1NI8C]
*Raimbert. French Literature in Mauritius (1800-1979): A Select Bibliography in the City Library of Port Louis. 1980 [https://books.google.co.uk/books?id=eQYpAQAAIAAJ]
*French Literature in Early American Translation: A Bibliographical Survey of Books and Pamphlets Printed in the United States from 1668 Through 1820. 1977. [https://books.google.co.uk/books?id=CYI0AQAAIAAJ]

History:
*Cambridge History of French Literature
*Coward. A History of French Literature: From Chanson de geste to Cinema. 2002. Paperback 2004. [https://books.google.co.uk/books?id=K8uS9vLpwuYC&amp;lpg=PP1&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Hollier. A New History of French Literature. 1989. 1994. [https://books.google.co.uk/books?id=nGQOodBVG9YC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Cazamian. A History of French Literature. Clarendon Press.1955. Reprinted 1967. [https://books.google.co.uk/books?id=W5PwAAAAMAAJ]
*Nitze and Dargan. A History of French Literature: From the Earliest Times to the Present. 1930. [https://books.google.co.uk/books?id=QT0nAAAAMAAJ]
*Butler. A History of French Literature. 1923. Reissued 1966. [https://books.google.co.uk/books?id=sPAoAAAAYAAJ]
*Dowden. A History of French Literature. (Literatures of the World). 1900 [https://books.google.co.uk/books?id=XWdcAAAAMAAJ]
*Wright. A History of French Literature. (Oxford French series). [https://books.google.co.uk/books?id=_O0oAAAAYAAJ]
*Schwarz. An Outline History of French Literature. 1924. 1932. [https://books.google.co.uk/books?id=9_fnAAAAMAAJ]
*Demogeot. History of French Literature. Adapted from the French by Bridge. 1874. [https://books.google.co.uk/books?id=LgxONYxlEogC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]

*Brunetière. Manual of the History of French Literature. 1898. [https://books.google.co.uk/books?id=WOHnAAAAMAAJ]

*Brereton. A Short History of French Literature. 1954. 2nd Ed:1976. [https://books.google.co.uk/books?id=DegoAAAAYAAJ]
*Bisson. A Short History of French Literature: From the Middle Ages to the Present Day. 1943. [https://books.google.co.uk/books?id=TLQywgEACAAJ]
*Hudson and Jack. A Short History of French Literature. 1919 [https://books.google.co.uk/books?id=8xIPAAAAQAAJ]
*Saintsbury. A Short History of French Literature. 1882. 2nd Ed: 1884. 7th Ed: 1917 [https://books.google.co.uk/books?id=WDoTAAAAMAAJ] [https://books.google.co.uk/books?id=AWFcAAAAMAAJ]

*Finch. French Literature: A Cultural History. 2010. [https://books.google.co.uk/books?id=8L0Z8uYUWj0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Prendergast. History of Modern French Literature: From the Sixteenth Century to the Twentieth Century. 2017. [https://books.google.co.uk/books?id=2QtpDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Laun. History of French Literature: From the Classical Renaissance until the End of the Reign of Louis XIV. 1883. [https://books.google.co.uk/books?id=bdZEAQAAIAAJ]

*Farrant. Introduction to Nineteenth-Century French Literature. 2007. [https://books.google.co.uk/books?id=m4HjBAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Vinet. History of French Literature in the Eighteenth Century. Translated by Bryce. 1854. [https://books.google.co.uk/books?id=5M803wEuBswC&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Jacqueline Cerquiglini-Toulet. A New History of Medieval French Literature. Translated by Sara Preisig. 2011. [https://books.google.co.uk/books?id=h42MQw6TfAcC&amp;pg=PT3#v=onepage&amp;q&amp;f=false]

*Konta. The History of French Literature: From the Oath of Strasburg to Chanticler. 1914. [https://books.google.co.uk/books?id=g2FcAAAAMAAJ]

*Cambridge Companion to Medieval French Literature

*Holmes. A History of Old French Literature: From the origins to 1300. 1938. Revised Ed: 1962. [https://books.google.co.uk/books?id=0UxcAAAAMAAJ]

*Sainstbury. A History of the French Novel (to the Close of the 19th Century). 1919. [https://books.google.co.uk/books?id=GAxJAQAAIAAJ]

==Japanese==
*Ryōzō Matsumoto. Japanese Literature New and Old. Hokuseido Press. 1966. [https://books.google.co.uk/books?id=EzftWTtGXzgC]
*J Ingram Bryan. The Literature of Japan. 1929. Kennikat Press. Port Washington. Reissued 1970. [https://books.google.co.uk/books?id=BwazAAAAIAAJ]
*Clay MacCauley. Japanese Literature. 1898. [https://books.google.co.uk/books?id=9nYuAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Epiphanius Wilson. Japanese Literature: Including Selections from Genji Monogatari and Classical Poetry and Drama of Japan. 1900. [https://books.google.co.uk/books?id=Iy4NAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Katsuhiko Takeda. Essays on Japanese Literature. Waseda University Press. 1977. [https://books.google.co.uk/books?id=VKkPAAAAYAAJ]

Bibliography
*Modern Japanese Literature in Western Translations: A Bibliography. International House of Japan Library. [https://books.google.co.uk/books?id=WZ7QAAAAMAAJ]

Periodicals
*Japanese Literature Today [https://books.google.co.uk/books?id=NyYHAQAAIAAJ]

Kokubungaku and nihonbungaku
*Tomoko Aoyama. &quot;From national literature to multicultural literature in Japanese language&quot;. Kaori Okano and Yoshio Sugimoto (eds). Rethinking Japanese Studies: Eurocentrism and the Asia-Pacific Region. Routledge Contemporary Japan Series. Chapter 4. pp [https://books.google.co.uk/books?id=sEcrDwAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false 53] to 72.

Reviewed
*Donald Richie. Japanese Literature Reviewed. 2003. [https://books.google.co.uk/books?id=ejJmAAAAMAAJ]

History
*Shuichi Kato. A History of Japanese Literature: From the Man'yōshū to Modern Times. New Abridged Edition. Japan Library. 1997. [https://books.google.co.uk/books?id=wUxOuD0NS5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Edward Putzar. Japanese Literature: A Historical Outline. University of Arizona Press. [https://books.google.co.uk/books?id=2kiBAAAAIAAJ]
*W G Aston. A History of Japanese Literature. (Literatures of the World). 1903. [https://books.google.co.uk/books?id=T4EMAAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Naomi Fukuda (ed). Literature. (Japanese History: A Guide to Survey Histories, Part 2). Center for Japanese Studies, University of Michigan. 1984. [https://books.google.co.uk/books?id=QsoUAQAAIAAJ]

Contemporary
*Kokusai Bunka Shinkokai (Japan Cultural Society). Introduction to Contemporary Japanese Literature: Synopese of Major Works: 1956-1970. University of Tokyo Press. 1972. [https://books.google.co.uk/books?id=XPQkHx6wR4IC]

Modern
*A Classified Catalogue of Modern Japanese Literature (Meiji, Taisho, Showa). University of Washington Library. 1964. [https://books.google.co.uk/books?id=lD8XAQAAMAAJ]
*Routledge Handbook of Modern Japanese Literature. 2016. [https://books.google.co.uk/books?id=EMpJDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*J Thomas Rimer and Van C Gessel (eds). The Columbia Anthology of Modern Japanese Literature. 2007. vol 2. [https://books.google.co.uk/books?id=BAg9tUJR1aEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
**Abridged. 2011. [https://books.google.co.uk/books?id=VrEYtVFv67oC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1].
*Karatani Kōjin. Origins of Modern Japanese Literature. 1993. [https://books.google.co.uk/books?id=hPJO2vEQgjYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Kan Kikuchi. History and Trends of Modern Japanese Literature. Tokyo. 1936. [https://books.google.co.uk/books?id=TU4DAAAAMAAJ]

Meiji and Taisho
*Kimura Ki (editor and compiler). Japanese Literature: Manners and Customs in the Meiji-Taishó Era. Ōbunsha. [https://books.google.co.uk/books?id=rBQrAAAAIAAJ]

Early modern
*Haruo Shirane (ed). Early Modern Japanese Literature: An Anthology, 1600-1900. Abridged Edition. 2008. [https://books.google.co.uk/books?id=SN72QCVBpVAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Classical
*The Princeton Companion to Classical Japanese Literature [https://books.google.co.uk/books?id=BSmMbQhafJoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Introduction to Classic Japanese Literature. Kokusai Bunka Shinkokai. 1948. [https://books.google.co.uk/books?id=WJAPAAAAYAAJ]

Traditional
*Haruo Shirane (ed). Traditional Japanese Literature: An Anthology, Beginnings to 1600. 2007. [https://books.google.co.uk/books?id=LsHfIsIXgEgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Literary criticism; Literary studies
*Seth Jacobowitz and Jonathan E Abel (eds). Modern Japanese Literary Studies. 2026. [https://books.google.co.uk/books?id=s-_AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Studies in Modern Japanese Literature: Essays and Translations in Honor of Edwin McClellan. 1997. [https://books.google.co.uk/books?id=_5EPAAAAYAAJ]

Anthology
*Donald Keene. Anthology of Japanese Literature from the earliest era to the mid-nineteenth century. Grove Press. 1955. [https://books.google.co.uk/books?id=9h8tAAAAMAAJ] Great Britain. 1956. [https://books.google.co.uk/books?id=LCxkAAAAMAAJ]. Evergreen Ed: 1960. [https://books.google.co.uk/books?id=rumBAAAAIAAJ]

[[Category:Literature]]</text>
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      <text bytes="14359" sha1="iwyproq2w0gbt3ss0z3jg7e8k5pclur" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Bibliography|Bibliography]]

This part of the [[Universal Bibliography]] is a bibliography of literature.

See [[s:Category:Literature]], [[s:Category:History of literature]], [[w:Bibliography of encyclopedias: literature]] and [[w:Category:Works about literature]]

==World==
*Cassell's Encyclopaedia of World Literature. 1953. 2nd Ed: 1973. vol 1 (histories and general articles): [https://books.google.co.uk/books?id=soIYAAAAIAAJ]. vol 3: [https://books.google.co.uk/books?id=AIEYAAAAIAAJ]. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA197#v=onepage&amp;q&amp;f=false].

*Damrosch. What is World Literature? 2003. [https://books.google.co.uk/books?id=yY-17mtp9R8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series:
*Edmund Gosse (ed). Literatures of the World. D Appleton and Co [https://en.m.wikisource.org/wiki/Page:A_history_of_Bohemian_literature.pdf/444]

==English==
*Bateson. Cambridge Bibliography of English Literature. 1940. Watson, Supplement 1957.
*Concise Cambridge Bibliography of English Literature
*New Cambridge Bibliography of English Literature
*Annual Bibliography of English Language and Literature. Cambridge University
*Ewen. Bibliography of Eighteenth Century English Literature. Columbia University. 1935
*Marcuse. A Reference Guide for English Studies. University of Calfornia. 1990
*Bracken. Reference Works in British and American Literature. Libraries Unlimited. 1998
*Kennedy and Sands. A Concise Bibliography for Students of English. Stanford University
*Pelican Guide to English Literature
*[[w:The Cambridge History of English and American Literature|Cambridge History of English and American Literature]]
*[[s:The Cambridge History of English Literature|Cambridge History of English Literature]]
*Concise Cambridge History of English Literature
*New Cambridge History of English Literature
*Oxford History of English Literature
*Oxford Illustrated History of English Literature
*Short Oxford History of English Literature
*Routledge History of Literature in English
*Cambridge History of Early Medieval English Literature
*Cambridge History of Medieval English Literature
*Cambridge History of Early Modern English Literature
*Cambridge History of Victorian Literature
*Cambridge History of Twentieth Century English Literature
*[[s:The Cambridge History of American Literature|Cambridge History of American Literature]]

Periodicals
*Liverpool Magazine (1890)
*Literary Garner (1835)

===United States===
See [[w:Category:American literature by state]]
*Dershem. An Outline of American State Literature. 1921

Arizona:
*Joseph Amasa Munk, History of Arizona Literature, 1925
*Mary G Boyer, Arizona in Literature, 1935
*Etulain, &quot;Contours of Culture in Arizona and the Modern West&quot; in Luey and Stowe, Arizona at  Seventy-five, 1987
*Diaz, &quot;A Bibliography of Bibliographies Relating to the History and Literature of Arizona and New Mexico&quot; (1958) 14 Arizona Quarterly 197

Colorado:
*Eugene Parsons, &quot;Colorado Literature&quot; in Stone, History of Colorado, 1918, volume 1, chapter 42, p 877 [https://books.google.co.uk/books?id=-uVYAAAAMAAJ]
*Levette J Davidson, &quot;The Literature of Colorado&quot; in Hafen. Colorado and Its People: A Narrative and Topical History of the Centennial State. Volume 2. Chapter 8. Page 225 [https://books.google.co.uk/books?id=sncXAAAAIAAJ]
*Fritz. &quot;Literature&quot;. Colorado, the Centennial State. 1941. p 417 [https://books.google.co.uk/books?id=EU0UAAAAYAAJ]
*&quot;Books and Writers&quot;. Colorado, a Guide to the Highest State. 1941. p 96 [https://books.google.co.uk/books?id=3o8GAQAAIAAJ]
*Eugene Parsons, &quot;The Study of Colorado Literature&quot; (1918) Colorado School Journal, vols 34-35, p 24 [https://books.google.co.uk/books?id=QxNRAQAAMAAJ]
*&quot;A Plea for the Study of Colorado Literature&quot; (1918) The Trail: A Magazine &quot;for Colorado&quot;, vol 11, p 12 [https://books.google.co.uk/books?id=K7QTAAAAYAAJ]

Oregon:
*Powers, Alfred. History of Oregon Literature. Metropolitan Press. 1935 [https://books.google.co.uk/books?id=JzELAAAAMAAJ]
*Horner, John B. Oregon Literature. 1899 [https://books.google.co.uk/books?id=3ecWSpoFMcQC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==French==
See [[s:Category:French literature]]

Bibliographies and bibliographical works:
*A Critical Bibliography of French Literature. Syracuse University Press. [https://books.google.co.uk/books?id=IFJQl7eUrg4C&amp;pg=PR3#v=onepage&amp;q&amp;f=false vol 5]
*Bassan, Breed and Spinelli. An Annotated Bibliography of French Language and Literature. 2nd Ed: 1976 [https://books.google.co.uk/books?id=BdkaAAAAMAAJ]
*French XX Bibliography: A Bibliography for the Study of French Literature and Culture since 1885. [https://books.google.co.uk/books?id=VxVxFxyDOmkC&amp;pg=PA19125#v=onepage&amp;q&amp;f=false]
*Foulet. A Bibliography of Medieval French Literature for College Libraries. 1915 [https://books.google.co.uk/books?id=httNAQAAIAAJ]
*Kirsop. The Bibliography of French Literary History: Progress, Problems, Projects. 1964. [https://books.google.co.uk/books?id=Xt7nAAAAMAAJ]

*Jaffe. Bibliography of French Literature in American Magazines in the 18th Century. Michigan State College Press. 1951. [https://books.google.co.uk/books?id=Cy7GGsr1NI8C]
*Raimbert. French Literature in Mauritius (1800-1979): A Select Bibliography in the City Library of Port Louis. 1980 [https://books.google.co.uk/books?id=eQYpAQAAIAAJ]
*French Literature in Early American Translation: A Bibliographical Survey of Books and Pamphlets Printed in the United States from 1668 Through 1820. 1977. [https://books.google.co.uk/books?id=CYI0AQAAIAAJ]

History:
*Cambridge History of French Literature
*Coward. A History of French Literature: From Chanson de geste to Cinema. 2002. Paperback 2004. [https://books.google.co.uk/books?id=K8uS9vLpwuYC&amp;lpg=PP1&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Hollier. A New History of French Literature. 1989. 1994. [https://books.google.co.uk/books?id=nGQOodBVG9YC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Cazamian. A History of French Literature. Clarendon Press.1955. Reprinted 1967. [https://books.google.co.uk/books?id=W5PwAAAAMAAJ]
*Nitze and Dargan. A History of French Literature: From the Earliest Times to the Present. 1930. [https://books.google.co.uk/books?id=QT0nAAAAMAAJ]
*Butler. A History of French Literature. 1923. Reissued 1966. [https://books.google.co.uk/books?id=sPAoAAAAYAAJ]
*Dowden. A History of French Literature. (Literatures of the World). 1900 [https://books.google.co.uk/books?id=XWdcAAAAMAAJ]
*Wright. A History of French Literature. (Oxford French series). [https://books.google.co.uk/books?id=_O0oAAAAYAAJ]
*Schwarz. An Outline History of French Literature. 1924. 1932. [https://books.google.co.uk/books?id=9_fnAAAAMAAJ]
*Demogeot. History of French Literature. Adapted from the French by Bridge. 1874. [https://books.google.co.uk/books?id=LgxONYxlEogC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]

*Brunetière. Manual of the History of French Literature. 1898. [https://books.google.co.uk/books?id=WOHnAAAAMAAJ]

*Brereton. A Short History of French Literature. 1954. 2nd Ed:1976. [https://books.google.co.uk/books?id=DegoAAAAYAAJ]
*Bisson. A Short History of French Literature: From the Middle Ages to the Present Day. 1943. [https://books.google.co.uk/books?id=TLQywgEACAAJ]
*Hudson and Jack. A Short History of French Literature. 1919 [https://books.google.co.uk/books?id=8xIPAAAAQAAJ]
*Saintsbury. A Short History of French Literature. 1882. 2nd Ed: 1884. 7th Ed: 1917 [https://books.google.co.uk/books?id=WDoTAAAAMAAJ] [https://books.google.co.uk/books?id=AWFcAAAAMAAJ]

*Finch. French Literature: A Cultural History. 2010. [https://books.google.co.uk/books?id=8L0Z8uYUWj0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Prendergast. History of Modern French Literature: From the Sixteenth Century to the Twentieth Century. 2017. [https://books.google.co.uk/books?id=2QtpDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Laun. History of French Literature: From the Classical Renaissance until the End of the Reign of Louis XIV. 1883. [https://books.google.co.uk/books?id=bdZEAQAAIAAJ]

*Farrant. Introduction to Nineteenth-Century French Literature. 2007. [https://books.google.co.uk/books?id=m4HjBAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Vinet. History of French Literature in the Eighteenth Century. Translated by Bryce. 1854. [https://books.google.co.uk/books?id=5M803wEuBswC&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Jacqueline Cerquiglini-Toulet. A New History of Medieval French Literature. Translated by Sara Preisig. 2011. [https://books.google.co.uk/books?id=h42MQw6TfAcC&amp;pg=PT3#v=onepage&amp;q&amp;f=false]

*Konta. The History of French Literature: From the Oath of Strasburg to Chanticler. 1914. [https://books.google.co.uk/books?id=g2FcAAAAMAAJ]

*Cambridge Companion to Medieval French Literature

*Holmes. A History of Old French Literature: From the origins to 1300. 1938. Revised Ed: 1962. [https://books.google.co.uk/books?id=0UxcAAAAMAAJ]

*Sainstbury. A History of the French Novel (to the Close of the 19th Century). 1919. [https://books.google.co.uk/books?id=GAxJAQAAIAAJ]

==Japanese==
*Ryōzō Matsumoto. Japanese Literature New and Old. Hokuseido Press. 1966. [https://books.google.co.uk/books?id=EzftWTtGXzgC]
*J Ingram Bryan. The Literature of Japan. 1929. Kennikat Press. Port Washington. Reissued 1970. [https://books.google.co.uk/books?id=BwazAAAAIAAJ]
*Clay MacCauley. Japanese Literature. 1898. [https://books.google.co.uk/books?id=9nYuAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Epiphanius Wilson. Japanese Literature: Including Selections from Genji Monogatari and Classical Poetry and Drama of Japan. 1900. [https://books.google.co.uk/books?id=Iy4NAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Katsuhiko Takeda. Essays on Japanese Literature. Waseda University Press. 1977. [https://books.google.co.uk/books?id=VKkPAAAAYAAJ]

Bibliography
*Modern Japanese Literature in Western Translations: A Bibliography. International House of Japan Library. [https://books.google.co.uk/books?id=WZ7QAAAAMAAJ]

Periodicals
*Japanese Literature Today [https://books.google.co.uk/books?id=NyYHAQAAIAAJ]

Kokubungaku and nihonbungaku
*Tomoko Aoyama. &quot;From national literature to multicultural literature in Japanese language&quot;. Kaori Okano and Yoshio Sugimoto (eds). Rethinking Japanese Studies: Eurocentrism and the Asia-Pacific Region. Routledge Contemporary Japan Series. Chapter 4. pp [https://books.google.co.uk/books?id=sEcrDwAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false 53] to 72.

Reviewed
*Donald Richie. Japanese Literature Reviewed. 2003. [https://books.google.co.uk/books?id=ejJmAAAAMAAJ]

History
*Shuichi Kato. A History of Japanese Literature: From the Man'yōshū to Modern Times. New Abridged Edition. Japan Library. 1997. [https://books.google.co.uk/books?id=wUxOuD0NS5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Edward Putzar. Japanese Literature: A Historical Outline. University of Arizona Press. [https://books.google.co.uk/books?id=2kiBAAAAIAAJ]
*W G Aston. A History of Japanese Literature. (Literatures of the World). 1903. [https://books.google.co.uk/books?id=T4EMAAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Naomi Fukuda (ed). Literature. (Japanese History: A Guide to Survey Histories, Part 2). Center for Japanese Studies, University of Michigan. 1984. [https://books.google.co.uk/books?id=QsoUAQAAIAAJ]

Contemporary
*Kokusai Bunka Shinkokai (Japan Cultural Society). Introduction to Contemporary Japanese Literature: Synopese of Major Works: 1956-1970. University of Tokyo Press. 1972. [https://books.google.co.uk/books?id=XPQkHx6wR4IC]

Modern
*A Classified Catalogue of Modern Japanese Literature (Meiji, Taisho, Showa). University of Washington Library. 1964. [https://books.google.co.uk/books?id=lD8XAQAAMAAJ]
*Routledge Handbook of Modern Japanese Literature. 2016. [https://books.google.co.uk/books?id=EMpJDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*J Thomas Rimer and Van C Gessel (eds). The Columbia Anthology of Modern Japanese Literature. 2007. vol 2. [https://books.google.co.uk/books?id=BAg9tUJR1aEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
**Abridged. 2011. [https://books.google.co.uk/books?id=VrEYtVFv67oC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1].
*Karatani Kōjin. Origins of Modern Japanese Literature. 1993. [https://books.google.co.uk/books?id=hPJO2vEQgjYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Kan Kikuchi. History and Trends of Modern Japanese Literature. Tokyo. 1936. [https://books.google.co.uk/books?id=TU4DAAAAMAAJ]
*Edward Mack. Manufacturing Modern Japanese Literature: Publishing, Prizes, and the Ascription of Literary Value. [https://books.google.co.uk/books?id=WnDI6s9surUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Donald Keene (ed). Modern Japanese Literature: From 1868 to the Present Day. Grove Press. 1956. [https://books.google.co.uk/books?id=5yxkAAAAMAAJ]
*Saeki Shōichi. Hidden Dimensions in Modern Japanese Literature. The Japan Foundation, Office for the Japanese Studies Center. [https://books.google.co.uk/books?id=OCYHAQAAIAAJ]

Meiji and Taisho
*Kimura Ki (editor and compiler). Japanese Literature: Manners and Customs in the Meiji-Taishó Era. Ōbunsha. [https://books.google.co.uk/books?id=rBQrAAAAIAAJ]

Early modern
*Haruo Shirane (ed). Early Modern Japanese Literature: An Anthology, 1600-1900. Abridged Edition. 2008. [https://books.google.co.uk/books?id=SN72QCVBpVAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Classical
*The Princeton Companion to Classical Japanese Literature [https://books.google.co.uk/books?id=BSmMbQhafJoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Introduction to Classic Japanese Literature. Kokusai Bunka Shinkokai. 1948. [https://books.google.co.uk/books?id=WJAPAAAAYAAJ]

Traditional
*Haruo Shirane (ed). Traditional Japanese Literature: An Anthology, Beginnings to 1600. 2007. [https://books.google.co.uk/books?id=LsHfIsIXgEgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Literary criticism; Literary studies
*Seth Jacobowitz and Jonathan E Abel (eds). Modern Japanese Literary Studies. 2026. [https://books.google.co.uk/books?id=s-_AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Studies in Modern Japanese Literature: Essays and Translations in Honor of Edwin McClellan. 1997. [https://books.google.co.uk/books?id=_5EPAAAAYAAJ]

Anthology
*Donald Keene. Anthology of Japanese Literature from the earliest era to the mid-nineteenth century. Grove Press. 1955. [https://books.google.co.uk/books?id=9h8tAAAAMAAJ] Great Britain. 1956. [https://books.google.co.uk/books?id=LCxkAAAAMAAJ]. Evergreen Ed: 1960. [https://books.google.co.uk/books?id=rumBAAAAIAAJ]

[[Category:Literature]]</text>
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      <text bytes="14482" sha1="9ef8io0gz3jeo7trz8mcf1hmm541zps" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Bibliography|Bibliography]]

This part of the [[Universal Bibliography]] is a bibliography of literature.

See [[s:Category:Literature]], [[s:Category:History of literature]], [[w:Bibliography of encyclopedias: literature]] and [[w:Category:Works about literature]]

==World==
*Cassell's Encyclopaedia of World Literature. 1953. 2nd Ed: 1973. vol 1 (histories and general articles): [https://books.google.co.uk/books?id=soIYAAAAIAAJ]. vol 3: [https://books.google.co.uk/books?id=AIEYAAAAIAAJ]. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA197#v=onepage&amp;q&amp;f=false].

*Damrosch. What is World Literature? 2003. [https://books.google.co.uk/books?id=yY-17mtp9R8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series:
*Edmund Gosse (ed). Literatures of the World. D Appleton and Co [https://en.m.wikisource.org/wiki/Page:A_history_of_Bohemian_literature.pdf/444]

==English==
*Bateson. Cambridge Bibliography of English Literature. 1940. Watson, Supplement 1957.
*Concise Cambridge Bibliography of English Literature
*New Cambridge Bibliography of English Literature
*Annual Bibliography of English Language and Literature. Cambridge University
*Ewen. Bibliography of Eighteenth Century English Literature. Columbia University. 1935
*Marcuse. A Reference Guide for English Studies. University of Calfornia. 1990
*Bracken. Reference Works in British and American Literature. Libraries Unlimited. 1998
*Kennedy and Sands. A Concise Bibliography for Students of English. Stanford University
*Pelican Guide to English Literature
*[[w:The Cambridge History of English and American Literature|Cambridge History of English and American Literature]]
*[[s:The Cambridge History of English Literature|Cambridge History of English Literature]]
*Concise Cambridge History of English Literature
*New Cambridge History of English Literature
*Oxford History of English Literature
*Oxford Illustrated History of English Literature
*Short Oxford History of English Literature
*Routledge History of Literature in English
*Cambridge History of Early Medieval English Literature
*Cambridge History of Medieval English Literature
*Cambridge History of Early Modern English Literature
*Cambridge History of Victorian Literature
*Cambridge History of Twentieth Century English Literature
*[[s:The Cambridge History of American Literature|Cambridge History of American Literature]]

Periodicals
*Liverpool Magazine (1890)
*Literary Garner (1835)

===United States===
See [[w:Category:American literature by state]]
*Dershem. An Outline of American State Literature. 1921

Arizona:
*Joseph Amasa Munk, History of Arizona Literature, 1925
*Mary G Boyer, Arizona in Literature, 1935
*Etulain, &quot;Contours of Culture in Arizona and the Modern West&quot; in Luey and Stowe, Arizona at  Seventy-five, 1987
*Diaz, &quot;A Bibliography of Bibliographies Relating to the History and Literature of Arizona and New Mexico&quot; (1958) 14 Arizona Quarterly 197

Colorado:
*Eugene Parsons, &quot;Colorado Literature&quot; in Stone, History of Colorado, 1918, volume 1, chapter 42, p 877 [https://books.google.co.uk/books?id=-uVYAAAAMAAJ]
*Levette J Davidson, &quot;The Literature of Colorado&quot; in Hafen. Colorado and Its People: A Narrative and Topical History of the Centennial State. Volume 2. Chapter 8. Page 225 [https://books.google.co.uk/books?id=sncXAAAAIAAJ]
*Fritz. &quot;Literature&quot;. Colorado, the Centennial State. 1941. p 417 [https://books.google.co.uk/books?id=EU0UAAAAYAAJ]
*&quot;Books and Writers&quot;. Colorado, a Guide to the Highest State. 1941. p 96 [https://books.google.co.uk/books?id=3o8GAQAAIAAJ]
*Eugene Parsons, &quot;The Study of Colorado Literature&quot; (1918) Colorado School Journal, vols 34-35, p 24 [https://books.google.co.uk/books?id=QxNRAQAAMAAJ]
*&quot;A Plea for the Study of Colorado Literature&quot; (1918) The Trail: A Magazine &quot;for Colorado&quot;, vol 11, p 12 [https://books.google.co.uk/books?id=K7QTAAAAYAAJ]

Oregon:
*Powers, Alfred. History of Oregon Literature. Metropolitan Press. 1935 [https://books.google.co.uk/books?id=JzELAAAAMAAJ]
*Horner, John B. Oregon Literature. 1899 [https://books.google.co.uk/books?id=3ecWSpoFMcQC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==French==
See [[s:Category:French literature]]

Bibliographies and bibliographical works:
*A Critical Bibliography of French Literature. Syracuse University Press. [https://books.google.co.uk/books?id=IFJQl7eUrg4C&amp;pg=PR3#v=onepage&amp;q&amp;f=false vol 5]
*Bassan, Breed and Spinelli. An Annotated Bibliography of French Language and Literature. 2nd Ed: 1976 [https://books.google.co.uk/books?id=BdkaAAAAMAAJ]
*French XX Bibliography: A Bibliography for the Study of French Literature and Culture since 1885. [https://books.google.co.uk/books?id=VxVxFxyDOmkC&amp;pg=PA19125#v=onepage&amp;q&amp;f=false]
*Foulet. A Bibliography of Medieval French Literature for College Libraries. 1915 [https://books.google.co.uk/books?id=httNAQAAIAAJ]
*Kirsop. The Bibliography of French Literary History: Progress, Problems, Projects. 1964. [https://books.google.co.uk/books?id=Xt7nAAAAMAAJ]

*Jaffe. Bibliography of French Literature in American Magazines in the 18th Century. Michigan State College Press. 1951. [https://books.google.co.uk/books?id=Cy7GGsr1NI8C]
*Raimbert. French Literature in Mauritius (1800-1979): A Select Bibliography in the City Library of Port Louis. 1980 [https://books.google.co.uk/books?id=eQYpAQAAIAAJ]
*French Literature in Early American Translation: A Bibliographical Survey of Books and Pamphlets Printed in the United States from 1668 Through 1820. 1977. [https://books.google.co.uk/books?id=CYI0AQAAIAAJ]

History:
*Cambridge History of French Literature
*Coward. A History of French Literature: From Chanson de geste to Cinema. 2002. Paperback 2004. [https://books.google.co.uk/books?id=K8uS9vLpwuYC&amp;lpg=PP1&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Hollier. A New History of French Literature. 1989. 1994. [https://books.google.co.uk/books?id=nGQOodBVG9YC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Cazamian. A History of French Literature. Clarendon Press.1955. Reprinted 1967. [https://books.google.co.uk/books?id=W5PwAAAAMAAJ]
*Nitze and Dargan. A History of French Literature: From the Earliest Times to the Present. 1930. [https://books.google.co.uk/books?id=QT0nAAAAMAAJ]
*Butler. A History of French Literature. 1923. Reissued 1966. [https://books.google.co.uk/books?id=sPAoAAAAYAAJ]
*Dowden. A History of French Literature. (Literatures of the World). 1900 [https://books.google.co.uk/books?id=XWdcAAAAMAAJ]
*Wright. A History of French Literature. (Oxford French series). [https://books.google.co.uk/books?id=_O0oAAAAYAAJ]
*Schwarz. An Outline History of French Literature. 1924. 1932. [https://books.google.co.uk/books?id=9_fnAAAAMAAJ]
*Demogeot. History of French Literature. Adapted from the French by Bridge. 1874. [https://books.google.co.uk/books?id=LgxONYxlEogC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]

*Brunetière. Manual of the History of French Literature. 1898. [https://books.google.co.uk/books?id=WOHnAAAAMAAJ]

*Brereton. A Short History of French Literature. 1954. 2nd Ed:1976. [https://books.google.co.uk/books?id=DegoAAAAYAAJ]
*Bisson. A Short History of French Literature: From the Middle Ages to the Present Day. 1943. [https://books.google.co.uk/books?id=TLQywgEACAAJ]
*Hudson and Jack. A Short History of French Literature. 1919 [https://books.google.co.uk/books?id=8xIPAAAAQAAJ]
*Saintsbury. A Short History of French Literature. 1882. 2nd Ed: 1884. 7th Ed: 1917 [https://books.google.co.uk/books?id=WDoTAAAAMAAJ] [https://books.google.co.uk/books?id=AWFcAAAAMAAJ]

*Finch. French Literature: A Cultural History. 2010. [https://books.google.co.uk/books?id=8L0Z8uYUWj0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Prendergast. History of Modern French Literature: From the Sixteenth Century to the Twentieth Century. 2017. [https://books.google.co.uk/books?id=2QtpDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Laun. History of French Literature: From the Classical Renaissance until the End of the Reign of Louis XIV. 1883. [https://books.google.co.uk/books?id=bdZEAQAAIAAJ]

*Farrant. Introduction to Nineteenth-Century French Literature. 2007. [https://books.google.co.uk/books?id=m4HjBAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Vinet. History of French Literature in the Eighteenth Century. Translated by Bryce. 1854. [https://books.google.co.uk/books?id=5M803wEuBswC&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Jacqueline Cerquiglini-Toulet. A New History of Medieval French Literature. Translated by Sara Preisig. 2011. [https://books.google.co.uk/books?id=h42MQw6TfAcC&amp;pg=PT3#v=onepage&amp;q&amp;f=false]

*Konta. The History of French Literature: From the Oath of Strasburg to Chanticler. 1914. [https://books.google.co.uk/books?id=g2FcAAAAMAAJ]

*Cambridge Companion to Medieval French Literature

*Holmes. A History of Old French Literature: From the origins to 1300. 1938. Revised Ed: 1962. [https://books.google.co.uk/books?id=0UxcAAAAMAAJ]

*Sainstbury. A History of the French Novel (to the Close of the 19th Century). 1919. [https://books.google.co.uk/books?id=GAxJAQAAIAAJ]

==Japanese==
*Ryōzō Matsumoto. Japanese Literature New and Old. Hokuseido Press. 1966. [https://books.google.co.uk/books?id=EzftWTtGXzgC]
*J Ingram Bryan. The Literature of Japan. 1929. Kennikat Press. Port Washington. Reissued 1970. [https://books.google.co.uk/books?id=BwazAAAAIAAJ]
*Clay MacCauley. Japanese Literature. 1898. [https://books.google.co.uk/books?id=9nYuAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Epiphanius Wilson. Japanese Literature: Including Selections from Genji Monogatari and Classical Poetry and Drama of Japan. 1900. [https://books.google.co.uk/books?id=Iy4NAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Katsuhiko Takeda. Essays on Japanese Literature. Waseda University Press. 1977. [https://books.google.co.uk/books?id=VKkPAAAAYAAJ]

Bibliography
*Modern Japanese Literature in Western Translations: A Bibliography. International House of Japan Library. [https://books.google.co.uk/books?id=WZ7QAAAAMAAJ]

Periodicals
*Japanese Literature Today [https://books.google.co.uk/books?id=NyYHAQAAIAAJ]

Kokubungaku and nihonbungaku
*Tomoko Aoyama. &quot;From national literature to multicultural literature in Japanese language&quot;. Kaori Okano and Yoshio Sugimoto (eds). Rethinking Japanese Studies: Eurocentrism and the Asia-Pacific Region. Routledge Contemporary Japan Series. Chapter 4. pp [https://books.google.co.uk/books?id=sEcrDwAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false 53] to 72.

Reviewed
*Donald Richie. Japanese Literature Reviewed. 2003. [https://books.google.co.uk/books?id=ejJmAAAAMAAJ]

History
*Shuichi Kato. A History of Japanese Literature: From the Man'yōshū to Modern Times. New Abridged Edition. Japan Library. 1997. [https://books.google.co.uk/books?id=wUxOuD0NS5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Edward Putzar. Japanese Literature: A Historical Outline. University of Arizona Press. [https://books.google.co.uk/books?id=2kiBAAAAIAAJ]
*W G Aston. A History of Japanese Literature. (Literatures of the World). 1903. [https://books.google.co.uk/books?id=T4EMAAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Naomi Fukuda (ed). Literature. (Japanese History: A Guide to Survey Histories, Part 2). Center for Japanese Studies, University of Michigan. 1984. [https://books.google.co.uk/books?id=QsoUAQAAIAAJ]

Today
*A Survey of Japanese Literature Today. Japan P.E.N. Club. 1984. [https://books.google.co.uk/books?id=OcFDAQAAIAAJ]

Contemporary
*Kokusai Bunka Shinkokai (Japan Cultural Society). Introduction to Contemporary Japanese Literature: Synopese of Major Works: 1956-1970. University of Tokyo Press. 1972. [https://books.google.co.uk/books?id=XPQkHx6wR4IC]

Modern
*A Classified Catalogue of Modern Japanese Literature (Meiji, Taisho, Showa). University of Washington Library. 1964. [https://books.google.co.uk/books?id=lD8XAQAAMAAJ]
*Routledge Handbook of Modern Japanese Literature. 2016. [https://books.google.co.uk/books?id=EMpJDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*J Thomas Rimer and Van C Gessel (eds). The Columbia Anthology of Modern Japanese Literature. 2007. vol 2. [https://books.google.co.uk/books?id=BAg9tUJR1aEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
**Abridged. 2011. [https://books.google.co.uk/books?id=VrEYtVFv67oC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1].
*Karatani Kōjin. Origins of Modern Japanese Literature. 1993. [https://books.google.co.uk/books?id=hPJO2vEQgjYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Kan Kikuchi. History and Trends of Modern Japanese Literature. Tokyo. 1936. [https://books.google.co.uk/books?id=TU4DAAAAMAAJ]
*Edward Mack. Manufacturing Modern Japanese Literature: Publishing, Prizes, and the Ascription of Literary Value. [https://books.google.co.uk/books?id=WnDI6s9surUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Donald Keene (ed). Modern Japanese Literature: From 1868 to the Present Day. Grove Press. 1956. [https://books.google.co.uk/books?id=5yxkAAAAMAAJ]
*Saeki Shōichi. Hidden Dimensions in Modern Japanese Literature. The Japan Foundation, Office for the Japanese Studies Center. [https://books.google.co.uk/books?id=OCYHAQAAIAAJ]

Meiji and Taisho
*Kimura Ki (editor and compiler). Japanese Literature: Manners and Customs in the Meiji-Taishó Era. Ōbunsha. [https://books.google.co.uk/books?id=rBQrAAAAIAAJ]

Early modern
*Haruo Shirane (ed). Early Modern Japanese Literature: An Anthology, 1600-1900. Abridged Edition. 2008. [https://books.google.co.uk/books?id=SN72QCVBpVAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Classical
*The Princeton Companion to Classical Japanese Literature [https://books.google.co.uk/books?id=BSmMbQhafJoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Introduction to Classic Japanese Literature. Kokusai Bunka Shinkokai. 1948. [https://books.google.co.uk/books?id=WJAPAAAAYAAJ]

Traditional
*Haruo Shirane (ed). Traditional Japanese Literature: An Anthology, Beginnings to 1600. 2007. [https://books.google.co.uk/books?id=LsHfIsIXgEgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Literary criticism; Literary studies
*Seth Jacobowitz and Jonathan E Abel (eds). Modern Japanese Literary Studies. 2026. [https://books.google.co.uk/books?id=s-_AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Studies in Modern Japanese Literature: Essays and Translations in Honor of Edwin McClellan. 1997. [https://books.google.co.uk/books?id=_5EPAAAAYAAJ]

Anthology
*Donald Keene. Anthology of Japanese Literature from the earliest era to the mid-nineteenth century. Grove Press. 1955. [https://books.google.co.uk/books?id=9h8tAAAAMAAJ] Great Britain. 1956. [https://books.google.co.uk/books?id=LCxkAAAAMAAJ]. Evergreen Ed: 1960. [https://books.google.co.uk/books?id=rumBAAAAIAAJ]

[[Category:Literature]]</text>
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      <text bytes="15323" sha1="6cx8z4a53234lmqtfwfcvmb04wzlqcf" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Bibliography|Bibliography]]

This part of the [[Universal Bibliography]] is a bibliography of literature.

See [[s:Category:Literature]], [[s:Category:History of literature]], [[w:Bibliography of encyclopedias: literature]] and [[w:Category:Works about literature]]

==World==
*Cassell's Encyclopaedia of World Literature. 1953. 2nd Ed: 1973. vol 1 (histories and general articles): [https://books.google.co.uk/books?id=soIYAAAAIAAJ]. vol 3: [https://books.google.co.uk/books?id=AIEYAAAAIAAJ]. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA197#v=onepage&amp;q&amp;f=false].

*Damrosch. What is World Literature? 2003. [https://books.google.co.uk/books?id=yY-17mtp9R8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series:
*Edmund Gosse (ed). Literatures of the World. D Appleton and Co [https://en.m.wikisource.org/wiki/Page:A_history_of_Bohemian_literature.pdf/444]

==English==
*Bateson. Cambridge Bibliography of English Literature. 1940. Watson, Supplement 1957.
*Concise Cambridge Bibliography of English Literature
*New Cambridge Bibliography of English Literature
*Annual Bibliography of English Language and Literature. Cambridge University
*Ewen. Bibliography of Eighteenth Century English Literature. Columbia University. 1935
*Marcuse. A Reference Guide for English Studies. University of Calfornia. 1990
*Bracken. Reference Works in British and American Literature. Libraries Unlimited. 1998
*Kennedy and Sands. A Concise Bibliography for Students of English. Stanford University
*Pelican Guide to English Literature
*[[w:The Cambridge History of English and American Literature|Cambridge History of English and American Literature]]
*[[s:The Cambridge History of English Literature|Cambridge History of English Literature]]
*Concise Cambridge History of English Literature
*New Cambridge History of English Literature
*Oxford History of English Literature
*Oxford Illustrated History of English Literature
*Short Oxford History of English Literature
*Routledge History of Literature in English
*Cambridge History of Early Medieval English Literature
*Cambridge History of Medieval English Literature
*Cambridge History of Early Modern English Literature
*Cambridge History of Victorian Literature
*Cambridge History of Twentieth Century English Literature
*[[s:The Cambridge History of American Literature|Cambridge History of American Literature]]

Periodicals
*Liverpool Magazine (1890)
*Literary Garner (1835)

===United States===
See [[w:Category:American literature by state]]
*Dershem. An Outline of American State Literature. 1921

Arizona:
*Joseph Amasa Munk, History of Arizona Literature, 1925
*Mary G Boyer, Arizona in Literature, 1935
*Etulain, &quot;Contours of Culture in Arizona and the Modern West&quot; in Luey and Stowe, Arizona at  Seventy-five, 1987
*Diaz, &quot;A Bibliography of Bibliographies Relating to the History and Literature of Arizona and New Mexico&quot; (1958) 14 Arizona Quarterly 197

Colorado:
*Eugene Parsons, &quot;Colorado Literature&quot; in Stone, History of Colorado, 1918, volume 1, chapter 42, p 877 [https://books.google.co.uk/books?id=-uVYAAAAMAAJ]
*Levette J Davidson, &quot;The Literature of Colorado&quot; in Hafen. Colorado and Its People: A Narrative and Topical History of the Centennial State. Volume 2. Chapter 8. Page 225 [https://books.google.co.uk/books?id=sncXAAAAIAAJ]
*Fritz. &quot;Literature&quot;. Colorado, the Centennial State. 1941. p 417 [https://books.google.co.uk/books?id=EU0UAAAAYAAJ]
*&quot;Books and Writers&quot;. Colorado, a Guide to the Highest State. 1941. p 96 [https://books.google.co.uk/books?id=3o8GAQAAIAAJ]
*Eugene Parsons, &quot;The Study of Colorado Literature&quot; (1918) Colorado School Journal, vols 34-35, p 24 [https://books.google.co.uk/books?id=QxNRAQAAMAAJ]
*&quot;A Plea for the Study of Colorado Literature&quot; (1918) The Trail: A Magazine &quot;for Colorado&quot;, vol 11, p 12 [https://books.google.co.uk/books?id=K7QTAAAAYAAJ]

Oregon:
*Powers, Alfred. History of Oregon Literature. Metropolitan Press. 1935 [https://books.google.co.uk/books?id=JzELAAAAMAAJ]
*Horner, John B. Oregon Literature. 1899 [https://books.google.co.uk/books?id=3ecWSpoFMcQC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==French==
See [[s:Category:French literature]]

Bibliographies and bibliographical works:
*A Critical Bibliography of French Literature. Syracuse University Press. [https://books.google.co.uk/books?id=IFJQl7eUrg4C&amp;pg=PR3#v=onepage&amp;q&amp;f=false vol 5]
*Bassan, Breed and Spinelli. An Annotated Bibliography of French Language and Literature. 2nd Ed: 1976 [https://books.google.co.uk/books?id=BdkaAAAAMAAJ]
*French XX Bibliography: A Bibliography for the Study of French Literature and Culture since 1885. [https://books.google.co.uk/books?id=VxVxFxyDOmkC&amp;pg=PA19125#v=onepage&amp;q&amp;f=false]
*Foulet. A Bibliography of Medieval French Literature for College Libraries. 1915 [https://books.google.co.uk/books?id=httNAQAAIAAJ]
*Kirsop. The Bibliography of French Literary History: Progress, Problems, Projects. 1964. [https://books.google.co.uk/books?id=Xt7nAAAAMAAJ]

*Jaffe. Bibliography of French Literature in American Magazines in the 18th Century. Michigan State College Press. 1951. [https://books.google.co.uk/books?id=Cy7GGsr1NI8C]
*Raimbert. French Literature in Mauritius (1800-1979): A Select Bibliography in the City Library of Port Louis. 1980 [https://books.google.co.uk/books?id=eQYpAQAAIAAJ]
*French Literature in Early American Translation: A Bibliographical Survey of Books and Pamphlets Printed in the United States from 1668 Through 1820. 1977. [https://books.google.co.uk/books?id=CYI0AQAAIAAJ]

History:
*Cambridge History of French Literature
*Coward. A History of French Literature: From Chanson de geste to Cinema. 2002. Paperback 2004. [https://books.google.co.uk/books?id=K8uS9vLpwuYC&amp;lpg=PP1&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Hollier. A New History of French Literature. 1989. 1994. [https://books.google.co.uk/books?id=nGQOodBVG9YC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Cazamian. A History of French Literature. Clarendon Press.1955. Reprinted 1967. [https://books.google.co.uk/books?id=W5PwAAAAMAAJ]
*Nitze and Dargan. A History of French Literature: From the Earliest Times to the Present. 1930. [https://books.google.co.uk/books?id=QT0nAAAAMAAJ]
*Butler. A History of French Literature. 1923. Reissued 1966. [https://books.google.co.uk/books?id=sPAoAAAAYAAJ]
*Dowden. A History of French Literature. (Literatures of the World). 1900 [https://books.google.co.uk/books?id=XWdcAAAAMAAJ]
*Wright. A History of French Literature. (Oxford French series). [https://books.google.co.uk/books?id=_O0oAAAAYAAJ]
*Schwarz. An Outline History of French Literature. 1924. 1932. [https://books.google.co.uk/books?id=9_fnAAAAMAAJ]
*Demogeot. History of French Literature. Adapted from the French by Bridge. 1874. [https://books.google.co.uk/books?id=LgxONYxlEogC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]

*Brunetière. Manual of the History of French Literature. 1898. [https://books.google.co.uk/books?id=WOHnAAAAMAAJ]

*Brereton. A Short History of French Literature. 1954. 2nd Ed:1976. [https://books.google.co.uk/books?id=DegoAAAAYAAJ]
*Bisson. A Short History of French Literature: From the Middle Ages to the Present Day. 1943. [https://books.google.co.uk/books?id=TLQywgEACAAJ]
*Hudson and Jack. A Short History of French Literature. 1919 [https://books.google.co.uk/books?id=8xIPAAAAQAAJ]
*Saintsbury. A Short History of French Literature. 1882. 2nd Ed: 1884. 7th Ed: 1917 [https://books.google.co.uk/books?id=WDoTAAAAMAAJ] [https://books.google.co.uk/books?id=AWFcAAAAMAAJ]

*Finch. French Literature: A Cultural History. 2010. [https://books.google.co.uk/books?id=8L0Z8uYUWj0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Prendergast. History of Modern French Literature: From the Sixteenth Century to the Twentieth Century. 2017. [https://books.google.co.uk/books?id=2QtpDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Laun. History of French Literature: From the Classical Renaissance until the End of the Reign of Louis XIV. 1883. [https://books.google.co.uk/books?id=bdZEAQAAIAAJ]

*Farrant. Introduction to Nineteenth-Century French Literature. 2007. [https://books.google.co.uk/books?id=m4HjBAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Vinet. History of French Literature in the Eighteenth Century. Translated by Bryce. 1854. [https://books.google.co.uk/books?id=5M803wEuBswC&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Jacqueline Cerquiglini-Toulet. A New History of Medieval French Literature. Translated by Sara Preisig. 2011. [https://books.google.co.uk/books?id=h42MQw6TfAcC&amp;pg=PT3#v=onepage&amp;q&amp;f=false]

*Konta. The History of French Literature: From the Oath of Strasburg to Chanticler. 1914. [https://books.google.co.uk/books?id=g2FcAAAAMAAJ]

*Cambridge Companion to Medieval French Literature

*Holmes. A History of Old French Literature: From the origins to 1300. 1938. Revised Ed: 1962. [https://books.google.co.uk/books?id=0UxcAAAAMAAJ]

*Sainstbury. A History of the French Novel (to the Close of the 19th Century). 1919. [https://books.google.co.uk/books?id=GAxJAQAAIAAJ]

==Japanese==
*Ryōzō Matsumoto. Japanese Literature New and Old. Hokuseido Press. 1966. [https://books.google.co.uk/books?id=EzftWTtGXzgC]
*J Ingram Bryan. The Literature of Japan. 1929. Kennikat Press. Port Washington. Reissued 1970. [https://books.google.co.uk/books?id=BwazAAAAIAAJ]
*Clay MacCauley. Japanese Literature. 1898. [https://books.google.co.uk/books?id=9nYuAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Epiphanius Wilson. Japanese Literature: Including Selections from Genji Monogatari and Classical Poetry and Drama of Japan. 1900. [https://books.google.co.uk/books?id=Iy4NAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Katsuhiko Takeda. Essays on Japanese Literature. Waseda University Press. 1977. [https://books.google.co.uk/books?id=VKkPAAAAYAAJ]

Bibliography
*Modern Japanese Literature in Western Translations: A Bibliography. International House of Japan Library. [https://books.google.co.uk/books?id=WZ7QAAAAMAAJ]

Periodicals
*Japanese Literature Today [https://books.google.co.uk/books?id=NyYHAQAAIAAJ]

Kokubungaku and nihonbungaku
*Tomoko Aoyama. &quot;From national literature to multicultural literature in Japanese language&quot;. Kaori Okano and Yoshio Sugimoto (eds). Rethinking Japanese Studies: Eurocentrism and the Asia-Pacific Region. Routledge Contemporary Japan Series. Chapter 4. pp [https://books.google.co.uk/books?id=sEcrDwAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false 53] to 72.

Reviewed
*Donald Richie. Japanese Literature Reviewed. 2003. [https://books.google.co.uk/books?id=ejJmAAAAMAAJ]

History
*Shuichi Kato. A History of Japanese Literature: From the Man'yōshū to Modern Times. New Abridged Edition. Japan Library. 1997. [https://books.google.co.uk/books?id=wUxOuD0NS5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Edward Putzar. Japanese Literature: A Historical Outline. University of Arizona Press. [https://books.google.co.uk/books?id=2kiBAAAAIAAJ]
*W G Aston. A History of Japanese Literature. (Literatures of the World). 1903. [https://books.google.co.uk/books?id=T4EMAAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Naomi Fukuda (ed). Literature. (Japanese History: A Guide to Survey Histories, Part 2). Center for Japanese Studies, University of Michigan. 1984. [https://books.google.co.uk/books?id=QsoUAQAAIAAJ]

Today
*A Survey of Japanese Literature Today. Japan P.E.N. Club. 1984. [https://books.google.co.uk/books?id=OcFDAQAAIAAJ]

Contemporary
*Kokusai Bunka Shinkokai (Japan Cultural Society). Introduction to Contemporary Japanese Literature: Synopese of Major Works: 1956-1970. University of Tokyo Press. 1972. [https://books.google.co.uk/books?id=XPQkHx6wR4IC]

Modern
*A Classified Catalogue of Modern Japanese Literature (Meiji, Taisho, Showa). University of Washington Library. 1964. [https://books.google.co.uk/books?id=lD8XAQAAMAAJ]
*Routledge Handbook of Modern Japanese Literature. 2016. [https://books.google.co.uk/books?id=EMpJDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*J Thomas Rimer and Van C Gessel (eds). The Columbia Anthology of Modern Japanese Literature. 2007. vol 2. [https://books.google.co.uk/books?id=BAg9tUJR1aEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
**Abridged. 2011. [https://books.google.co.uk/books?id=VrEYtVFv67oC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1].
*Karatani Kōjin. Origins of Modern Japanese Literature. 1993. [https://books.google.co.uk/books?id=hPJO2vEQgjYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Kan Kikuchi. History and Trends of Modern Japanese Literature. Tokyo. 1936. [https://books.google.co.uk/books?id=TU4DAAAAMAAJ]
*Edward Mack. Manufacturing Modern Japanese Literature: Publishing, Prizes, and the Ascription of Literary Value. [https://books.google.co.uk/books?id=WnDI6s9surUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Donald Keene (ed). Modern Japanese Literature: From 1868 to the Present Day. Grove Press. 1956. [https://books.google.co.uk/books?id=5yxkAAAAMAAJ]
*Saeki Shōichi. Hidden Dimensions in Modern Japanese Literature. The Japan Foundation, Office for the Japanese Studies Center. [https://books.google.co.uk/books?id=OCYHAQAAIAAJ]

Meiji and Taisho
*Kimura Ki (editor and compiler). Japanese Literature: Manners and Customs in the Meiji-Taishó Era. Ōbunsha. [https://books.google.co.uk/books?id=rBQrAAAAIAAJ]

Early modern
*Haruo Shirane (ed). Early Modern Japanese Literature: An Anthology, 1600-1900. Abridged Edition. 2008. [https://books.google.co.uk/books?id=SN72QCVBpVAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Classical
*The Princeton Companion to Classical Japanese Literature [https://books.google.co.uk/books?id=BSmMbQhafJoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Introduction to Classic Japanese Literature. Kokusai Bunka Shinkokai. 1948. [https://books.google.co.uk/books?id=WJAPAAAAYAAJ]

Traditional
*Haruo Shirane (ed). Traditional Japanese Literature: An Anthology, Beginnings to 1600. 2007. [https://books.google.co.uk/books?id=LsHfIsIXgEgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Literary criticism; Literary studies
*Seth Jacobowitz and Jonathan E Abel (eds). Modern Japanese Literary Studies. 2026. [https://books.google.co.uk/books?id=s-_AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Studies in Modern Japanese Literature: Essays and Translations in Honor of Edwin McClellan. 1997. [https://books.google.co.uk/books?id=_5EPAAAAYAAJ]

Anthology
*Donald Keene. Anthology of Japanese Literature from the earliest era to the mid-nineteenth century. Grove Press. 1955. [https://books.google.co.uk/books?id=9h8tAAAAMAAJ] Great Britain. 1956. [https://books.google.co.uk/books?id=LCxkAAAAMAAJ]. Evergreen Ed: 1960. [https://books.google.co.uk/books?id=rumBAAAAIAAJ]

Novels

Modern novels
*Nancy Junko Beauchamp. Modern Japanese Novels in English: A Selected Bibliography. Service Center for Teachers of Asian Studies, Association for Asian Studies, Ohio State University. May 1974. [https://books.google.co.uk/books?id=WltEAAAAIAAJ]
*Kinya Tsuruta and Thomas E Swann (eds). Approaches to the Modern Japanese Novel. Sophia University. Tokyo. 1976. [https://books.google.co.uk/books?id=bhdkAAAAMAAJ]

Short stories

Modern short stories
*Thomas E Swann and Kinya Tsuruta (eds). Approaches to the Modern Japanese Short Story. Waseda University Press. 1982. [https://books.google.co.uk/books?id=IJYPAAAAYAAJ]

Poetry

Modern poetry
*A R Davis (ed). Modern Japanese Poetry. University of Queensland Press. 1978. The Open University Press. Milton Keynes. 1979. [https://books.google.co.uk/books?id=v0aBAAAAIAAJ]

[[Category:Literature]]</text>
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  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
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      <timestamp>2026-08-04T13:59:44Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
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      <text bytes="9878" sha1="qkwcm5qgth82mhrv03m1nqpbmuzfkjl" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260803.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
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    <title>Motivation and emotion/Assessment/Using generative AI</title>
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        <id>10242</id>
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__TOC__
&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
[[File:Astrocyte (NIH BioArt 40 - 627579).png|80px|right]]
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Permitted]''' - The ethical use of GenAI is allowed in completing the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].
* '''[https://www.canberra.edu.au/about-uc/l-and-t/genai-and-assessment-at-uc Restricted]''' - The use of GenAI is NOT allowed in completing the [[Motivation and emotion/Assessment/Exam|exam]].

To use genAI appropriately:
* Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] with a publicly accessible link to the conversation or tool used and prompt details.
* [[w:Fact-checking|Fact-check]] genAI content.
* Only cite peer-reviewed sources which you have consulted.
* Human-rewrite genAI content to enhance quality.&lt;includeonly&gt;
* [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]].

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for using the Cogniti [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU book chapter topic generator],  Studiosity Writing Feedback+, or for low-level tasks such as improving spelling and grammar.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). These tools can include the Cogniti book chapter topic generator, Studiosity Writing Feedback+, and frontier models.

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use of the Cogniti book chapter topic generator, Studiosity Writing Feedback+, or for low-level tasks such as fixing grammar and spelling.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact-check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. [https://www.seangoedecke.com/llms-reward-expertise GenAI tools work best for topics which you already understand].

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

==External links==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

[[Category:Motivation and emotion/Assessment]]
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  <page>
    <title>Bully Metric Timestamps</title>
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      <text bytes="35044" sha1="tnxwtwea0gmazzefjffe1s32m7jxfcx" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|400px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 200
|cHeight = 200
|oTop = 0
|oLeft = 0
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 230
|cHeight = 230
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|400px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>tnxwtwea0gmazzefjffe1s32m7jxfcx</sha1>
    </revision>
    <revision>
      <id>2820587</id>
      <parentid>2820586</parentid>
      <timestamp>2026-08-04T17:18:01Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35044" sha1="g93sudgff2fl37b9lz72pf2tqik121p" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|400px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 180
|cHeight = 180
|oTop = 0
|oLeft = 0
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 200
|cHeight = 200
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|400px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>g93sudgff2fl37b9lz72pf2tqik121p</sha1>
    </revision>
    <revision>
      <id>2820588</id>
      <parentid>2820587</parentid>
      <timestamp>2026-08-04T17:19:40Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35044" sha1="7vs8n0dkyrfoodmdng6a09oeuua5nzn" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|380px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 160
|cHeight = 160
|oTop = 0
|oLeft = 0
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|380px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>7vs8n0dkyrfoodmdng6a09oeuua5nzn</sha1>
    </revision>
    <revision>
      <id>2820589</id>
      <parentid>2820588</parentid>
      <timestamp>2026-08-04T17:21:21Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35044" sha1="asez3yaqg0bjn908m07vxwmjq5jb9yx" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|350px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 140
|cHeight = 140
|oTop = 0
|oLeft = 0
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|350px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>asez3yaqg0bjn908m07vxwmjq5jb9yx</sha1>
    </revision>
    <revision>
      <id>2820590</id>
      <parentid>2820589</parentid>
      <timestamp>2026-08-04T17:22:42Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35046" sha1="o3ck7ybphweuggn6511b5p63pppj1dx" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|350px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 140
|cHeight = 140
|oTop = 50
|oLeft = 50
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|350px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>o3ck7ybphweuggn6511b5p63pppj1dx</sha1>
    </revision>
    <revision>
      <id>2820591</id>
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        <username>Unitfreak</username>
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      <text bytes="35046" sha1="dqukq9ew35ecl6vrxra8cam4z4u071v" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|350px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 130
|cHeight = 130
|oTop = 10
|oLeft = 50
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|350px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>dqukq9ew35ecl6vrxra8cam4z4u071v</sha1>
    </revision>
    <revision>
      <id>2820592</id>
      <parentid>2820591</parentid>
      <timestamp>2026-08-04T17:23:37Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35046" sha1="2s5wcifjvi50j7ugfo71t2nn12trgem" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|350px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 130
|cHeight = 130
|oTop = 15
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|350px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>2s5wcifjvi50j7ugfo71t2nn12trgem</sha1>
    </revision>
    <revision>
      <id>2820593</id>
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        <username>Unitfreak</username>
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      <text bytes="35046" sha1="0ly5ren90ax4bm42e47raqmpyktih6h" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|350px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 15
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|350px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>0ly5ren90ax4bm42e47raqmpyktih6h</sha1>
    </revision>
    <revision>
      <id>2820594</id>
      <parentid>2820593</parentid>
      <timestamp>2026-08-04T17:24:43Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820594</origin>
      <model>wikitext</model>
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      <text bytes="35046" sha1="3ig9nndd2a1h0n9e1md52kn3hf08sl8" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|350px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1800
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|350px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>3ig9nndd2a1h0n9e1md52kn3hf08sl8</sha1>
    </revision>
    <revision>
      <id>2820595</id>
      <parentid>2820594</parentid>
      <timestamp>2026-08-04T17:25:03Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35046" sha1="ec46nd8mld35dvmb6xxpqdguj1zja52" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|350px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 780
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|350px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>ec46nd8mld35dvmb6xxpqdguj1zja52</sha1>
    </revision>
    <revision>
      <id>2820596</id>
      <parentid>2820595</parentid>
      <timestamp>2026-08-04T17:25:28Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35046" sha1="lx6ml2ydkktohbz6a204hd7o9skh7b4" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|350px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|350px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>lx6ml2ydkktohbz6a204hd7o9skh7b4</sha1>
    </revision>
    <revision>
      <id>2820597</id>
      <parentid>2820596</parentid>
      <timestamp>2026-08-04T17:27:24Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35046" sha1="l2lt3wa8xzd96bfd99ebcbar33cwmqo" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>l2lt3wa8xzd96bfd99ebcbar33cwmqo</sha1>
    </revision>
    <revision>
      <id>2820601</id>
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      <timestamp>2026-08-04T20:52:04Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="36167" sha1="aowavwx0kincg8mvwzwnrvn0wp82b0n" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described previously, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:

:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;


 '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
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|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>aowavwx0kincg8mvwzwnrvn0wp82b0n</sha1>
    </revision>
    <revision>
      <id>2820602</id>
      <parentid>2820601</parentid>
      <timestamp>2026-08-04T20:52:23Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820602</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35831" sha1="da62njkfvspg5711iqrwt853ztbgybj" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described previously, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:

:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>da62njkfvspg5711iqrwt853ztbgybj</sha1>
    </revision>
    <revision>
      <id>2820603</id>
      <parentid>2820602</parentid>
      <timestamp>2026-08-04T20:53:07Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="35847" sha1="slbn9ze76dczlpoog243m7673pm4k8x" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described previously, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;
&amp;thinsp;
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>slbn9ze76dczlpoog243m7673pm4k8x</sha1>
    </revision>
    <revision>
      <id>2820604</id>
      <parentid>2820603</parentid>
      <timestamp>2026-08-04T21:15:31Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820604</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="36097" sha1="1sxoz12hoie0bej46wtu9chv5ybx7bt" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described previously, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;
&amp;thinsp;
Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be more precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>1sxoz12hoie0bej46wtu9chv5ybx7bt</sha1>
    </revision>
    <revision>
      <id>2820605</id>
      <parentid>2820604</parentid>
      <timestamp>2026-08-04T21:15:50Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820605</origin>
      <model>wikitext</model>
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      <text bytes="36106" sha1="by65oh9srcoism1r5m3x8kj40c30esu" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described previously, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;
&amp;thinsp;
&amp;thinsp;
Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be more precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>by65oh9srcoism1r5m3x8kj40c30esu</sha1>
    </revision>
    <revision>
      <id>2820606</id>
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        <username>Unitfreak</username>
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      <text bytes="36094" sha1="7edxvd9lwtetlb7bljda988z3iyhv9x" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described previously, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;
&lt;br/&gt;
Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be more precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>7edxvd9lwtetlb7bljda988z3iyhv9x</sha1>
    </revision>
    <revision>
      <id>2820607</id>
      <parentid>2820606</parentid>
      <timestamp>2026-08-04T21:16:26Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820607</origin>
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      <text bytes="36089" sha1="elhoe7vny0mfu9mz1ixy79xv4fcztlf" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described previously, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be more precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>elhoe7vny0mfu9mz1ixy79xv4fcztlf</sha1>
    </revision>
    <revision>
      <id>2820608</id>
      <parentid>2820607</parentid>
      <timestamp>2026-08-04T21:17:21Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="36084" sha1="kivyr25iotuuyhtt6h2te7bjxryxkpi" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be more precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>kivyr25iotuuyhtt6h2te7bjxryxkpi</sha1>
    </revision>
    <revision>
      <id>2820615</id>
      <parentid>2820608</parentid>
      <timestamp>2026-08-04T21:39:03Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="36633" sha1="7a911loou0rtyfjqrgsh1uy8sbe76a5" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|right|450px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 ''R''&lt;sub&gt;☉&lt;/sub&gt;, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this distance.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>7a911loou0rtyfjqrgsh1uy8sbe76a5</sha1>
    </revision>
    <revision>
      <id>2820617</id>
      <parentid>2820615</parentid>
      <timestamp>2026-08-04T21:46:07Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="36634" sha1="l1h1igduv8ms14ssnmq8c0xgpfeetgf" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 ''R''&lt;sub&gt;☉&lt;/sub&gt;, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this distance.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
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    </revision>
    <revision>
      <id>2820618</id>
      <parentid>2820617</parentid>
      <timestamp>2026-08-04T22:04:02Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820618</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="36655" sha1="i29gzkoixbopkb6rvitixwhckx8kjyx" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 ''R''&lt;sub&gt;☉&lt;/sub&gt;, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 3a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 3b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 3c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 3d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>i29gzkoixbopkb6rvitixwhckx8kjyx</sha1>
    </revision>
    <revision>
      <id>2820619</id>
      <parentid>2820618</parentid>
      <timestamp>2026-08-04T22:08:10Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="36690" sha1="p62f58n6q8h2tgrk3flyfu5g9262lev" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|right|450px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>p62f58n6q8h2tgrk3flyfu5g9262lev</sha1>
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      <id>2820620</id>
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      <text bytes="36691" sha1="j74hm3b0fiqq1bumj3n51xdt04krv20" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the time required for the sun to orbit one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents the time required for 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>j74hm3b0fiqq1bumj3n51xdt04krv20</sha1>
    </revision>
    <revision>
      <id>2820621</id>
      <parentid>2820620</parentid>
      <timestamp>2026-08-04T22:51:09Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="36647" sha1="cg9wgpdgr5q9adin9jt548iofhkioak" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first Bully timestamp digit (furthest right) represents the orbital motion of one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>cg9wgpdgr5q9adin9jt548iofhkioak</sha1>
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    <revision>
      <id>2820622</id>
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      <text bytes="36652" sha1="2bfbp45qr1jjnvitzrwxo92qjhqkgen" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first Bully timestamp digit (furthest right) represents the orbital time period of one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>2bfbp45qr1jjnvitzrwxo92qjhqkgen</sha1>
    </revision>
    <revision>
      <id>2820625</id>
      <parentid>2820622</parentid>
      <timestamp>2026-08-05T03:00:06Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="36651" sha1="fjmax1v8womaspohryhwc25a9a9q7yg" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>fjmax1v8womaspohryhwc25a9a9q7yg</sha1>
    </revision>
    <revision>
      <id>2820636</id>
      <parentid>2820625</parentid>
      <timestamp>2026-08-05T03:38:01Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="37324" sha1="exue8fvnsu52e7yf7vaxlk274bqmzqq" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these values are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated by the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>exue8fvnsu52e7yf7vaxlk274bqmzqq</sha1>
    </revision>
    <revision>
      <id>2820638</id>
      <parentid>2820636</parentid>
      <timestamp>2026-08-05T03:42:41Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820638</origin>
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      <text bytes="37327" sha1="32kb3tt5w6g2ftobc39tj4eujjs6kk6" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated by the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>32kb3tt5w6g2ftobc39tj4eujjs6kk6</sha1>
    </revision>
    <revision>
      <id>2820639</id>
      <parentid>2820638</parentid>
      <timestamp>2026-08-05T03:43:44Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="37398" sha1="4yehrrwc0xm3z72xws0t1rjo9zl2t9z" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated by the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>4yehrrwc0xm3z72xws0t1rjo9zl2t9z</sha1>
    </revision>
    <revision>
      <id>2820640</id>
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      <timestamp>2026-08-05T03:45:21Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="37398" sha1="f7lv8j3d371qwk3vylxcxfkea5yg6m0" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 3:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 3). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 3''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 4''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 4:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>f7lv8j3d371qwk3vylxcxfkea5yg6m0</sha1>
    </revision>
    <revision>
      <id>2820641</id>
      <parentid>2820640</parentid>
      <timestamp>2026-08-05T03:49:06Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* The Galactic Calendar */</comment>
      <origin>2820641</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="37403" sha1="7gf2ibx9s687mboeux7a13m8ghuuck8" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>7gf2ibx9s687mboeux7a13m8ghuuck8</sha1>
    </revision>
    <revision>
      <id>2820642</id>
      <parentid>2820641</parentid>
      <timestamp>2026-08-05T04:18:34Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820642</origin>
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      <format>text/x-wiki</format>
      <text bytes="39395" sha1="jn030uar76511ym6l17a2547vjterdb" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in Figure 3 are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>jn030uar76511ym6l17a2547vjterdb</sha1>
    </revision>
    <revision>
      <id>2820643</id>
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      <timestamp>2026-08-05T04:20:23Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="37404" sha1="6nzy5ww7yawdw7mq6z42hw40nroll7e" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]


{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>6nzy5ww7yawdw7mq6z42hw40nroll7e</sha1>
    </revision>
    <revision>
      <id>2820644</id>
      <parentid>2820643</parentid>
      <timestamp>2026-08-05T04:21:16Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */  This entire edit was provided by Google Gemini.</comment>
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      <text bytes="39395" sha1="jn030uar76511ym6l17a2547vjterdb" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in Figure 3 are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>jn030uar76511ym6l17a2547vjterdb</sha1>
    </revision>
    <revision>
      <id>2820645</id>
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      <timestamp>2026-08-05T04:37:25Z</timestamp>
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        <username>Unitfreak</username>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="39927" sha1="1okqneu41b8r4gnuntxb8isfw1az3yo" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' can be estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in Figure 3 are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
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|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>1okqneu41b8r4gnuntxb8isfw1az3yo</sha1>
    </revision>
    <revision>
      <id>2820646</id>
      <parentid>2820645</parentid>
      <timestamp>2026-08-05T04:53:22Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820646</origin>
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      <text bytes="39923" sha1="3ryu2pw0iiu4i1tq8oda8hfyxazlyga" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' is estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in Figure 3 are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>3ryu2pw0iiu4i1tq8oda8hfyxazlyga</sha1>
    </revision>
    <revision>
      <id>2820647</id>
      <parentid>2820646</parentid>
      <timestamp>2026-08-05T05:20:22Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="39937" sha1="q9v0eule9gonl88fwjv5n4gljgwypz6" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' is estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>q9v0eule9gonl88fwjv5n4gljgwypz6</sha1>
    </revision>
    <revision>
      <id>2820648</id>
      <parentid>2820647</parentid>
      <timestamp>2026-08-05T05:45:09Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
      <origin>2820648</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="39973" sha1="0q857xnj4wa0ejeyumjfy31pgi29gp4" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' is estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>0q857xnj4wa0ejeyumjfy31pgi29gp4</sha1>
    </revision>
    <revision>
      <id>2820649</id>
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        <username>Unitfreak</username>
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      <comment>/* The Pleiades star cluster */</comment>
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      <text bytes="40005" sha1="3d3sjw5p0gib6wop6k43fyabl1beeb2" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' is estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>3d3sjw5p0gib6wop6k43fyabl1beeb2</sha1>
    </revision>
    <revision>
      <id>2820650</id>
      <parentid>2820649</parentid>
      <timestamp>2026-08-05T05:55:46Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Galactic Weeks */</comment>
      <origin>2820650</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="40006" sha1="9syd9tqinu7bs0vijp70nxmc244fjqg" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' is estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of the '''1st Quarter, Galactic Week 0 (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
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      <id>2820651</id>
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      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Galactic Weeks */</comment>
      <origin>2820651</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="40010" sha1="9w40dl3z1nx1whz1567na6ze676zqj7" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. Timestamp '''8209 0000 0000''' is estimated to have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of '''Galactic Week 0 of the 1st Quarter,  (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>9w40dl3z1nx1whz1567na6ze676zqj7</sha1>
    </revision>
    <revision>
      <id>2820652</id>
      <parentid>2820651</parentid>
      <timestamp>2026-08-05T06:04:50Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="40021" sha1="4qrj3ixchpthn9cnknin8erchs3wek5" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of '''Galactic Week 0 of the 1st Quarter,  (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>4qrj3ixchpthn9cnknin8erchs3wek5</sha1>
    </revision>
    <revision>
      <id>2820660</id>
      <parentid>2820652</parentid>
      <timestamp>2026-08-05T06:19:43Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="40110" sha1="dof884bo4m28mc1hme7k84b0tqb76me" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (approximately 3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs, meaning that the appearance of the night sky will completely change over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of '''Galactic Week 0 of the 1st Quarter,  (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>dof884bo4m28mc1hme7k84b0tqb76me</sha1>
    </revision>
    <revision>
      <id>2820661</id>
      <parentid>2820660</parentid>
      <timestamp>2026-08-05T06:26:36Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="40110" sha1="j0ivjykc7a1fx8ewh2iukva5w11zlrp" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for approximately one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs, meaning that the appearance of the night sky will completely change over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
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|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
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|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
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[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of '''Galactic Week 0 of the 1st Quarter,  (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>j0ivjykc7a1fx8ewh2iukva5w11zlrp</sha1>
    </revision>
    <revision>
      <id>2820662</id>
      <parentid>2820661</parentid>
      <timestamp>2026-08-05T06:28:45Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <text bytes="40118" sha1="r4m4567lnmsgd17ju3oy3untourmzg2" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents the orbital time for approximately one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt; (3,055 seconds), and the fifth digit represents 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;, or roughly 6.344 years. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs, meaning that the appearance of the night sky will completely change over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of '''Galactic Week 0 of the 1st Quarter,  (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>r4m4567lnmsgd17ju3oy3untourmzg2</sha1>
    </revision>
    <revision>
      <id>2820663</id>
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      <timestamp>2026-08-05T06:33:03Z</timestamp>
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        <username>Unitfreak</username>
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      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="40164" sha1="2kymjw7n5hhl6yrx4wiprizbyez8y4t" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents 3,055 seconds, which is the orbital time for approximately one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt;. The fifth digit represents approximately 6.344 years, or roughly the time required to travel 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs, meaning that the appearance of the night sky will completely change over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of '''Galactic Week 0 of the 1st Quarter,  (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>2kymjw7n5hhl6yrx4wiprizbyez8y4t</sha1>
    </revision>
    <revision>
      <id>2820664</id>
      <parentid>2820663</parentid>
      <timestamp>2026-08-05T06:33:41Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Naked Eye Stars */</comment>
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      <text bytes="40163" sha1="98mkcjyje1ay8s1u1ywpgxm703p1998" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents 3,055 seconds, which is the orbital time of approximately one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt;. The fifth digit represents approximately 6.344 years, or roughly the time required to travel 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs, meaning that the appearance of the night sky will completely change over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over the time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of '''Galactic Week 0 of the 1st Quarter,  (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
      <sha1>98mkcjyje1ay8s1u1ywpgxm703p1998</sha1>
    </revision>
    <revision>
      <id>2820665</id>
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        <username>Unitfreak</username>
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      <text bytes="40161" sha1="414gvmc1mv5di6fb7a1wgkjv0pn5nnb" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by approximately one solar radius along its path through the Galaxy. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;
&amp;thinsp;
[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|right|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

The Sun orbits the center of the Milky Way galaxy at a very fast speed, roughly 227.7 kilometers per second (km/s), which equals approximately 0.076% of the speed of light. Even though the Sun is moving very quickly, it is also physically immense. The radius of the Sun (&lt;math&gt;R_\odot&lt;/math&gt;) is 695,700 kilometers. Dividing the solar radius by the galactic orbital velocity, we find that it takes approximately '''3055 seconds''' for the Sun to travel a distance equal to its own radius:
&amp;thinsp;
:&lt;math&gt;\Delta t = \frac{695,700 \text{ km}}{227.7 \text{ km/s}} \approx 3055 \text{ seconds}&lt;/math&gt;
&amp;thinsp;
'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. The sequential timestamp, '''8209 2800 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998''' (where TAI is International Atomic Time). As visually shown in Figure 1, the Sun orbited a distance of one solar radius during this 3,055 second time period.

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', the heliosphere is a vast, oblong, tailed, bubble-like region that extends from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere, except that Earth's atmosphere is a comparatively thin layer of gas that remains near the Earth's surface. By comparison, the heliosphere is a plasma that is constantly blasted out into space due to the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is very large. It is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; (65,536) solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately '''65,536 solar radii''', or roughly the diameter of one spherical heliosphere. Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. 

As explained previously, timestamp '''8209 2800 0000''' is defined to have occurred at exactly '''12:00:00 TAI on June 21, 1998'''. Timestamp '''8209 2801 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

=== Naked Eye Stars ===
As described above, the first digit (furthest right) in a Bully timestamp represents 3,055 seconds, which is the orbital time of approximately one solar radius ''R''&lt;sub&gt;☉&lt;/sub&gt;. The fifth digit represents approximately 6.344 years, or roughly the time required to travel 16&lt;sup&gt;4&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;. Before moving on to describe the physical significance of 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; in terms of &quot;naked eye stars&quot;, it is worth noting that the length 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is remarkably close to 10&lt;sup&gt;10&lt;/sup&gt; light-seconds. In fact, these distances are so similar (differing by less than 0.35%) that one can estimate the ratio of the sun's orbital speed to the speed of light by dividing:
&amp;thinsp;
:&lt;math&gt;\frac{10^{10}}{16^8 \times 3055} \approx 0.076\%&lt;/math&gt;

Furthermore, these values (16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; and 10&lt;sup&gt;10&lt;/sup&gt; light-seconds) are of the same order of magnitude as 100 parsecs, where a parsec (roughly 3.26 light-years) is a common length unit used in astronomy. To be precise, 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt; is approximately 96.83 parsecs.

'''Figure 3:''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred roughly sixty-three thousand BC, and timestamp '''820A 0000 0000''' is estimated to occur around three hundred and fifty-three thousand AD, for a total time lapse of four hundred and sixteen thousand years. The stacked histogram in Figure 3 has a red dashed line showing 96.83 parsecs (the distance the Sun will travel in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps). As indicated in the histogram, a large percentage of &quot;Naked Eye&quot; stars are nearer to the sun than 96.83 parsecs, meaning that the appearance of the night sky will completely change over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of &quot;Naked Eye&quot; stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 16^8 R_☉, which is the distance that the sun travels in 16^8 Bully timestamps.|'''Figure 3:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of &quot;Naked Eye&quot; stars are within this travel distance of the sun, 96.83 parsecs or 16&lt;sup&gt;8&lt;/sup&gt; ''R''&lt;sub&gt;☉&lt;/sub&gt;.]]

The term ''Naked Eye Stars'' refers to any celestial object that can be seen in the night sky using only human vision, completely unaided by binoculars or telescopes. However, what qualifies as a &quot;naked eye star&quot; is highly subjective, depending heavily on environmental light pollution and a person's biological visual acuity. In remote regions like deserts or high mountains, the sky is perfectly dark. A person may see up to 2,500 to 3,500 stars at a given time. The Milky Way can actually cast shadows on the ground in these conditions. In major metropolitan areas like New York or Tokyo, extreme light pollution blanks out the sky. Only the Moon, planets, and perhaps a dozen or two of the absolute brightest stars remain visible to the naked eye.

To see faint stars, human eyes must adapt to the dark, widening the pupils to draw in light. A young person's pupil may expand to 7 mm, whereas an older adult’s pupil might only expand to 5 mm, naturally making faint stars invisible to the older observer. Also, minor uncorrected astigmatisms, nearsightedness, or mild cataracts smudge pinpoint starlight, causing faint stars to blend directly into the background glow of the night sky.

In 129 B.C., the ancient Greek astronomer Hipparchus created the world's first stellar catalog. He ranked the stars purely by how they appeared to his naked eye. In 1856, astronomer Norman Pogson formalized this ancient system mathematically. He discovered that the human eye perceives brightness logarithmically, and that Hipparchus’s 1st magnitude stars were exactly 100 times brighter than his 6th magnitude stars.

* '''1st Magnitude:''' The very brightest, &quot;first-rate&quot; stars to light up at twilight.
* '''2nd, 3rd, 4th, 5th Magnitude:''' Progressively dimmer stars.
* '''6th Magnitude:''' The absolute faintest, &quot;sixth-rate&quot; stars Hipparchus could barely see under pristine, ancient night skies.

The stars in the Figure 3 histogram are ranked using the modern version of Hipparchus' &quot;Magnitude&quot; system. A total of 9,427 stars are included in the stacked histogram, but more than two thirds of these are 6th Magnitude stars that are only visible in ideal circumstances. It is notable that stars of first through third magnitude tend to be nearer than 100 parsecs, whereas stars of fifth and sixth magnitude tend to be beyond the 100 parsecs mark. Over a time duration of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps, the Sun will travel a distance that is beyond the majority of the brightest stars, but not as far as the dimmest Naked Eye stars.

==== The Pleiades star cluster ====
{| class=&quot;wikitable&quot; style=&quot;margin-left: auto; margin-right: auto; border: none; background: transparent;&quot;
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:Magnitude_illustration.svg|thumb|right|340px|alt=TBD.|'''Figure 4a:''' An SVG illustration of magnitude in astronomy.]]
|-
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 200
|cWidth = 120
|cHeight = 120
|oTop = 12
|oLeft = 40
|Location = left
|Description = '''Figure 4b:''' The combined apparent magnitude of the Pleiades star cluster (Messier 45) is approximately 1.6 when viewed together as a group.
}}
| style=&quot;border: none; padding: 10px;&quot; |
{{CSS image crop
|Image = Pleiades_over_Arizona.jpg
|bSize = 1700
|cWidth = 180
|cHeight = 180
|oTop = 500
|oLeft = 750
|Location = center
|Description = '''Figure 4c:''' The 9 star cluster is composed of 1 third-magnitude star, 5 fourth-magnitude stars, 2 fifth-magnitude stars, and 1 sixth magnitude star.
}}
|-
| colspan = 2; style=&quot;border: none; padding: 10px;&quot; |
[[File:M45map.jpg|thumb|right|340px|alt=TBD.|'''Figure 4d:''' TBD.]]

|}

== The Galactic Calendar ==
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the galactic center over a span of 250 million years.|'''Figure 5a:''' Stars orbiting around the Galactic center during a 250 million-year time period.]]
A '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits (see Figure 5a). Stars closer to the center will orbit much more quickly than those on the outer edges. The stars shown in '''Figure 5a''' all eventually localized near the Sun despite having vastly different historical orbital trajectories, visually illustrating the long-term uncertainty of galactic orbits.

=== Bully Galactic Years ===
If the Sun followed a perfectly circular orbit around the Milky Way, as estimated in the text in the lower right corner of '''Figure 5b''', the radius of that orbit would be approximately 26,000 light-years. The time required for the Sun to complete one full circular orbit would be calculated by dividing the orbital circumference by the orbital speed:
&amp;thinsp;
:&lt;math&gt;\begin{aligned}
\Delta t &amp;= \frac{2\pi \times 26,000 \text{ light-years}}{230 \text{ km/s}} \\
&amp;\approx 213 \text{ million years}
\end{aligned}&lt;/math&gt;
&amp;thinsp;
[[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|thumb|center|600px|alt=Diagram showing the intertwined orbital paths of the Earth and Moon as they accompany the Sun on its massive orbit around the Milky Way center.|'''Figure 5b:''' Motion of the Sun, Earth, and Moon around the Milky Way Galaxy.]]
&amp;thinsp;
Within the context of the Bully timekeeping system, a '''Bully galactic year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years). While this is not identical to a true, observed galactic year, it should be noted that the true long-term trajectory of the Sun is inherently chaotic and unpredictable over deep time. Therefore, this fixed power-of-two value serves as a reasonable approximation.

=== Bully Galactic Year 65 ===
Since the Bully system utilizes hexadecimal notation and a Bully Galactic Year spans 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamp intervals, the positional values of the highest digits map directly to large cosmic eras:

* The '''twelfth digit''' (the far-left position) scales in increments of 8 Bully Galactic Years:
**&lt;math&gt;\frac{16^{11}}{2^{41}} = 8&lt;/math&gt;.
* The '''eleventh digit''' scales in increments of half a Bully Galactic Year:
**&lt;math&gt;\frac{16^{10}}{2^{41}} = \frac{1}{2}&lt;/math&gt;.
* The '''tenth digit''' scales in increments of one-thirty-second of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{9}}{2^{41}} = \frac{1}{32}&lt;/math&gt;.
* The '''ninth digit''' scales in increments of one five-hundred-and-twelfth of a Bully Galactic Year:
**&lt;math&gt;\frac{16^{8}}{2^{41}} = \frac{1}{512}&lt;/math&gt;.

Any timestamp in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicates that the system is recording time within the '''65th Bully Galactic Year''' of the Universe.  However, the Sun (and our solar system) did not come into existence until approximately 44 Bully Galactic Years after the Big Bang, meaning our solar system is only '''21 Bully Galactic Years old'''.

=== Galactic Weeks ===
A '''Galactic Week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.92 degrees''' around the galactic center (approximately 4.1 million years), so that 52 Galactic Weeks is equivalent to one Galactic Year. The following table (see Figure 5) illustrates the division of one Galactic Year's worth of Bully timestamps into 52 equal portions. Galactic Year &quot;65&quot; begins with Bully timestamp '''8200 0000 0000''' and ends with timestamp '''83FF FFFF FFFF'''. We are currently nearing the end of '''Galactic Week 0 of the 1st Quarter,  (8200 0000 0000 - 8209 D89D 89D7)'''.

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 5: Bully Galactic Year 65

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Galactic &lt;br /&gt; Year 65
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}}
|}


* [[Bully_Metric_Astronomical_Coordinates|Learn More About Galactic Years and The Bully Metric Coordinate System]]

==== The Metonic Cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moon Metonic Cycles
* July 23, 1998 on 8209 280'''0 038B'''
* July 23, 2017 on 8209 280'''3 0238'''
* July 23, 2036 on 8209 280'''6 00EA'''
* July 23, 2055 on 8209 280'''8 FF9B'''
* July 23, 2074 on 8209 280'''B FE45'''
* July 23, 2093 on 8209 280'''E FCE6'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

=== Time Estimation Divisions ===
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Used to estimate cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 3 contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 3: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 4 (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 4: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 4) measure &quot;lookback&quot; time anchored at timestamp ''8209 2800 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 5 is the same as is shown in Figure 4, but Figure 5 plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 5: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6 is similar to the table in Figure 3, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 3 was for large z values, Figure 6 shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;”
| style=&quot;font-weight: bold; background-color: #eaecf0;&quot; | {{nowrap|8209 2800 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]

== The Bully Mnemonic ==

&lt;math display=&quot;block&quot;&gt; {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; {1 \, Tropical \, Year} = {31,556,926 \, Seconds} &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt; 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years &lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years &lt;/math&gt;


The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]
* [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|User Name}}
# [[/Expectancy–value theory of educational motivation/]] - What is expectancy–value theory and how can it be applied to understand and enhance educational motivation? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|U3280251}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|U3183521}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? - {{ME-By|U3222012}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|U3237996}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Adaptive versus maladaptive self-reflection/]] – When does self-reflection promote wellbeing and when does it contribute to psychological distress? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|U3285438}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Outdoor play and children's emotional well-being/]] - How does outdoor play influence children's emotional well-being? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book/2026]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|User Name}}
# [[/Expectancy–value theory of educational motivation/]] - What is expectancy–value theory and how can it be applied to understand and enhance educational motivation? {{ME-By|StudentUC2026}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|U3280251}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|U3183521}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? - {{ME-By|U3222012}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|U3237996}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Adaptive versus maladaptive self-reflection/]] – When does self-reflection promote wellbeing and when does it contribute to psychological distress? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|U3285438}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Outdoor play and children's emotional well-being/]] - How does outdoor play influence children's emotional well-being? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book/2026]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|User Name}}
# [[/Expectancy–value theory of educational motivation/]] - What is expectancy–value theory and how can it be applied to understand and enhance educational motivation? {{ME-By|StudentUC2026}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|u3242837}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|U3280251}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|U3183521}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? - {{ME-By|U3222012}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|U3237996}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Adaptive versus maladaptive self-reflection/]] – When does self-reflection promote wellbeing and when does it contribute to psychological distress? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|U3285438}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Outdoor play and children's emotional well-being/]] - How does outdoor play influence children's emotional well-being? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book/2026]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|User Name}}
# [[/Expectancy–value theory of educational motivation/]] - What is expectancy–value theory and how can it be applied to understand and enhance educational motivation? {{ME-By|StudentUC2026}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|U3242837}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|U3280251}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|U3183521}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? - {{ME-By|U3222012}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|U3237996}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Adaptive versus maladaptive self-reflection/]] – When does self-reflection promote wellbeing and when does it contribute to psychological distress? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|U3285438}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Outdoor play and children's emotional well-being/]] - How does outdoor play influence children's emotional well-being? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book/2026]]</text>
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  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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      <text bytes="88672" sha1="jjq2lgu8fbf1hqoevzyi6jfm8dxfl7f" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders which are actually [[w:SO(4)#Visualization_of_4D_rotations|bent into tori in 4-space]].    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°, and those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. The right and left rotations are enantiomorphous reflections of each other like a pair of clasped hands, with non-congruent vertex position sequences. The right rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. In the left rotation the roles of the short chord polygon and the long chord polygon are reversed: the short chord polygons remain stationary in 4-space as vertices circle over them, as the rotation takes Clifford parallel long chord polygons to each other.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <comment>Undid revision [[Special:Diff/2820578|2820578]] by [[Special:Contributions/Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|talk]])</comment>
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      <text bytes="88668" sha1="06g6nmgt6b81twp2apdb7y0fom9vaiu" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders which are actually [[w:SO(4)#Visualization_of_4D_rotations|bent into tori in 4-space]].    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. The right and left rotations are enantiomorphous reflections of each other like a pair of clasped hands, with non-congruent vertex position sequences. The right rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. In the left rotation the roles of the short chord polygon and the long chord polygon are reversed: the short chord polygons remain stationary in 4-space as vertices circle over them, as the rotation takes Clifford parallel long chord polygons to each other.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>06g6nmgt6b81twp2apdb7y0fom9vaiu</sha1>
    </revision>
    <revision>
      <id>2820580</id>
      <parentid>2820579</parentid>
      <timestamp>2026-08-04T16:24:55Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 24-cell */</comment>
      <origin>2820580</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="88664" sha1="3s2uq46ssc2fs5wgj51wep1cl5r3jso" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders which are actually [[w:SO(4)#Visualization_of_4D_rotations|bent into tori in 4-space]].    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. Right and left rotations are enantiomorphous reflections of each other like a pair of clasped hands, with non-congruent vertex position sequences. The right rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. In the left rotation the roles of the short chord polygon and the long chord polygon are reversed: the short chord polygons remain stationary in 4-space as vertices circle over them, as the rotation takes Clifford parallel long chord polygons to each other.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>3s2uq46ssc2fs5wgj51wep1cl5r3jso</sha1>
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    <revision>
      <id>2820581</id>
      <parentid>2820580</parentid>
      <timestamp>2026-08-04T16:29:41Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 24-cell */</comment>
      <origin>2820581</origin>
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      <text bytes="88678" sha1="g8yo0ud1somgegnhwggfl9xl6h6moon" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders which are actually [[w:SO(4)#Visualization_of_4D_rotations|bent into tori in 4-space]].    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, with non-congruent vertex position sequences, like a pair of clasped hands. The right rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. In the left rotation the roles of the short chord polygon and the long chord polygon are reversed. The short chord polygons remain stationary in 4-space as vertices circle over them, while the rotation takes Clifford parallel long chord polygons to each other.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="88680" sha1="mjcxduywkwm1ajceeswiryff9qshqkc" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - August 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders which are actually [[w:SO(4)#Visualization_of_4D_rotations|bent into tori in 4-space]].    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, with non-congruent vertex position sequences, like a pair of clasped hands. The right rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. In the left rotation the roles of the short chord polygon and the long chord polygon are reversed. The short chord polygons remain stationary in 4-space as vertices circle over them, while the rotation takes Clifford parallel long chord polygons to each other.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>mjcxduywkwm1ajceeswiryff9qshqkc</sha1>
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      <text bytes="88540" sha1="aq91e20v88btsdjq1g480zgsdizdtci" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - August 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders which are actually [[w:SO(4)#Visualization_of_4D_rotations|bent into tori in 4-space]].    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, with non-congruent vertex position sequences, like a pair of clasped hands. The right rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. In the left rotation the roles of the short chord polygon and the long chord polygon are reversed. The short chord polygons remain stationary in 4-space as vertices circle over them, while the rotation takes Clifford parallel long chord polygons to each other.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>aq91e20v88btsdjq1g480zgsdizdtci</sha1>
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      <id>2820584</id>
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      <text bytes="88569" sha1="f8pf4ahw38dfdl4vqablp5pvwy8wezn" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - August 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders which are actually [[w:SO(4)#Visualization_of_4D_rotations|bent into tori in 4-space]].    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, with non-congruent vertex position sequences, like a pair of clasped hands. The right rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. In the left rotation the roles of the short chord polygon and the long chord polygon are reversed. The short chord polygons remain stationary in 4-space as vertices circle over them, while the rotation takes Clifford parallel long chord polygons to each other.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>f8pf4ahw38dfdl4vqablp5pvwy8wezn</sha1>
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      <id>2820585</id>
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      <text bytes="88778" sha1="ak5cr3q0qr53p88khfvwxqqm3jr3xgg" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - August 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders which are actually [[w:SO(4)#Visualization_of_4D_rotations|bent into tori in 4-space]].    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, with non-congruent vertex position sequences, like a pair of clasped hands. The right rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. In the left rotation the roles of the short chord polygon and the long chord polygon are reversed. The short chord polygons remain stationary in 4-space as vertices circle over them, while the rotation takes Clifford parallel long chord polygons to each other.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>ak5cr3q0qr53p88khfvwxqqm3jr3xgg</sha1>
    </revision>
  </page>
  <page>
    <title>Mandelbrot CLI: Renderer with Perturbation Theory</title>
    <ns>0</ns>
    <id>330398</id>
    <revision>
      <id>2820598</id>
      <parentid>2818350</parentid>
      <timestamp>2026-08-04T18:02:43Z</timestamp>
      <contributor>
        <username>Aokoroko</username>
        <id>2811879</id>
      </contributor>
      <comment>/* C++ Source Code */</comment>
      <origin>2820598</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10697" sha1="0du3f4b59bh18h1q6079bwmcb7nihbe" xml:space="preserve">== Introduction ==
This page contains the original C++ source code used to render high-precision fragments of the Mandelbrot set using perturbation theory and 8x8 Super-Sampling Anti-Aliasing (SSAA). Created by [[User:Aokoroko]].

== Key Features ==
* '''High-Precision Reference:''' The 5000-bit reference trajectory is computed exactly once per zoom layer.
* '''Hardware-Native Performance:''' Blazing-fast math for billions of pixels utilizing hardware-native double registers.
* When using double-precision floating-point numbers (on the order of 10⁻¹⁵), perturbation theory only allows you to zoom down to the '''10⁻³⁰⁸ level—no further.'''
* '''Innovative Algorithm:''' Revolutionary *Reference Reset to Zero* implementation.
* '''True 8x8 SSAA:''' Pristine, anti-aliased image quality with 64 independent samples per pixel.
* '''OpenMP Multi-threading:''' High-speed parallel computing to maximize CPU utilization.

== C++ Source Code ==

&lt;syntaxhighlight lang=&quot;cpp&quot;&gt;
#include &lt;atomic&gt;
#include &lt;cmath&gt;
#include &lt;cstdint&gt;
#include &lt;cstdio&gt;
#include &lt;vector&gt;
#include &lt;algorithm&gt;
#include &lt;mpfr.h&gt;
#include &lt;omp.h&gt;
using std::vector;
using std::min;
const char * CENTER_RE = &quot;-1.99999543561201124623198345433951143502785679245726844745821388800402678499411681518036306219179273434395557574279985918047221291197081186140687781560831995&quot;;
const char * CENTER_IM = &quot;-0.00000000000000000000000026198152173811047783694060060607013913873144250985383083459221663448338433592617272786772587281530484110756597337683912309313885172&quot;;
const char * VIEW_SIZE = &quot;1.15e-119&quot;;
const int WIDTH = 2160;
const int HEIGHT = 2160;
const int AA = 8;
const int MAX_ITER = 50000;
const double ESCAPE_RADIUS_SQUARED = 50000.0;
const int PALETTE_FRAME = 200;
const char * OUTPUT_FILE = &quot;Mandelbrot Set Image 112.bmp&quot;;
const mpfr_prec_t PRECISION_BITS = 1000;
struct Complex {
  double re;
  double im;
};
#pragma pack(push, 1)
struct BMPHeader {
  uint16_t type{0x4D42};
  uint32_t size{0};
  uint32_t reserved{0};
  uint32_t offBits{54};
  uint32_t structSize{40};
  int32_t width{0};
  int32_t height{0};
  uint16_t planes{1};
  uint16_t bitCount{24};
  uint32_t compression{0};
  uint32_t sizeImage{0};
  int32_t xPixelsPerMeter{2834};
  int32_t yPixelsPerMeter{2834};
  uint32_t colorsUsed{0};
  uint32_t colorsImportant{0};
};
#pragma pack(pop)
int main() {
  const double startTime = omp_get_wtime();
  const long rawWidth = static_cast&lt;long&gt;(WIDTH) * AA;
  const long rawHeight = static_cast&lt;long&gt;(HEIGHT) * AA;
  mpfr_t centerRe, centerIm, zReMp, zImMp, tmp1, tmp2, viewSizeMp;
  mpfr_inits2(PRECISION_BITS, centerRe, centerIm, zReMp, zImMp, tmp1, tmp2, viewSizeMp, static_cast&lt;mpfr_ptr&gt;(nullptr));
  mpfr_set_str(centerRe, CENTER_RE, 10, MPFR_RNDN);
  mpfr_set_str(centerIm, CENTER_IM, 10, MPFR_RNDN);
  mpfr_set_str(viewSizeMp, VIEW_SIZE, 10, MPFR_RNDN);
  const double sampleStep = mpfr_get_d(viewSizeMp, MPFR_RNDN) / rawWidth;
  vector&lt;Complex&gt; referenceOrbit;
  referenceOrbit.reserve(MAX_ITER + 200);
  mpfr_set_ui(zReMp, 0, MPFR_RNDN);
  mpfr_set_ui(zImMp, 0, MPFR_RNDN);
  for (int iter = 0; iter &lt;= MAX_ITER + 150; ++iter) {
    Complex z{mpfr_get_d(zReMp, MPFR_RNDN), mpfr_get_d(zImMp, MPFR_RNDN)};
    referenceOrbit.push_back(z);
    if (z.re * z.re + z.im * z.im &gt; ESCAPE_RADIUS_SQUARED &amp;&amp; iter &gt; MAX_ITER) {
      break;
    }
    mpfr_mul(tmp1, zReMp, zImMp, MPFR_RNDN);
    mpfr_sqr(tmp2, zReMp, MPFR_RNDN);
    mpfr_sqr(zReMp, zImMp, MPFR_RNDN);
    mpfr_sub(zReMp, tmp2, zReMp, MPFR_RNDN);
    mpfr_add(zReMp, zReMp, centerRe, MPFR_RNDN);
    mpfr_mul_2ui(tmp1, tmp1, 1, MPFR_RNDN);
    mpfr_add(zImMp, tmp1, centerIm, MPFR_RNDN);
  }
  const int referenceLength = static_cast&lt;int&gt;(referenceOrbit.size());
  mpfr_clears(centerRe, centerIm, zReMp, zImMp, tmp1, tmp2, viewSizeMp, static_cast&lt;mpfr_ptr&gt;(nullptr));
  std::fprintf(stderr, &quot;Reference orbit: %d points\n&quot;, referenceLength);
  std::fprintf(stderr, &quot;Precomputing skip100 matrices...\n&quot;);
  vector&lt;Complex&gt; coeff_A(referenceLength, {1.0, 0.0});
  vector&lt;Complex&gt; coeff_B(referenceLength, {0.0, 0.0});
  vector&lt;double&gt; rad_R(referenceLength, 2.0);
  #pragma omp parallel for
  for (int i = 0; i &lt; MAX_ITER; ++i) {
    double min_r = 2.0;
    for (int k = 0; k &lt; 100; ++k) {
      if (i + k &gt;= referenceLength) break;
      double r_re = referenceOrbit[i + k].re;
      double r_im = referenceOrbit[i + k].im;
      double abs_s = std::sqrt(r_re * r_re + r_im * r_im);
      double aS = (abs_s &lt; 2.0) ? abs_s : 0.0;
      min_r = min(min_r, aS);
      double next_A_re = 2.0 * (r_re * coeff_A[i].re - r_im * coeff_A[i].im);
      double next_A_im = 2.0 * (r_re * coeff_A[i].im + r_im * coeff_A[i].re);
      double next_B_re = 2.0 * (r_re * coeff_B[i].re - r_im * coeff_B[i].im) + 1.0;
      double next_B_im = 2.0 * (r_re * coeff_B[i].im + r_im * coeff_B[i].re);
      coeff_A[i].re = next_A_re; coeff_A[i].im = next_A_im;
      coeff_B[i].re = next_B_re; coeff_B[i].im = next_B_im;
    }
    rad_R[i] = min_r;
  }
  const double PI = 3.14159265358979323846;
  uint8_t palette[256][3];
  for (int i = 0; i &lt; 255; ++i) {
    palette[i][0] = static_cast&lt;uint8_t&gt;(std::lround(127.0 + 127.0 * std::cos(2.0 * PI * i / 255.0)));
    palette[i][1] = static_cast&lt;uint8_t&gt;(std::lround(127.0 + 127.0 * std::sin(2.0 * PI * i / 255.0)));
    palette[i][2] = palette[i][1];
  }
  palette[255][0] = 255; palette[255][1] = 255; palette[255][2] = 255;
  const int rowBytes = (WIDTH * 3 + 3) &amp; ~3;
  vector&lt;uint8_t&gt; image(static_cast&lt;size_t&gt;(rowBytes) * HEIGHT, 0);
  std::atomic&lt;int&gt; completedRows{0};
  const Complex * reference = referenceOrbit.data();
  #pragma omp parallel for schedule(dynamic)
  for (int y = 0; y &lt; HEIGHT; ++y) {
    uint8_t * row = image.data() + static_cast&lt;size_t&gt;(y) * rowBytes;
    for (int x = 0; x &lt; WIDTH; ++x) {
      unsigned blueSum = 0; unsigned greenSum = 0; unsigned redSum = 0;
      for (int sampleY = 0; sampleY &lt; AA; ++sampleY) {
        const double deltaCIm = (static_cast&lt;long&gt;(y) * AA + sampleY - rawHeight / 2) * sampleStep;
        for (int sampleX = 0; sampleX &lt; AA; ++sampleX) {
          const double deltaCRe = (static_cast&lt;long&gt;(x) * AA + sampleX - rawWidth / 2) * sampleStep;
          double deltaRe = 0.0; double deltaIm = 0.0;
          double zRe = 0.0;     double zIm = 0.0;
          int referenceIndex = 0;
          int iter = 0;
while (iter &lt; MAX_ITER) {
    if (zRe * zRe + zIm * zIm &gt;= ESCAPE_RADIUS_SQUARED) {
        break;
    }
    double eps_abs2 = deltaRe * deltaRe + deltaIm * deltaIm;
    double limit_r2 = 1e-60 * rad_R[referenceIndex] * rad_R[referenceIndex];
    if (eps_abs2 &lt; limit_r2 &amp;&amp; (referenceIndex + 100 &lt; referenceLength - 1) &amp;&amp; (iter + 100 &lt; MAX_ITER)) {
        double backup_deltaRe = deltaRe; double backup_deltaIm = deltaIm;
        int backup_refIdx = referenceIndex; int backup_iter = iter;
        double next_eps_re = (coeff_A[referenceIndex].re * deltaRe - coeff_A[referenceIndex].im * deltaIm) + 
                             (coeff_B[referenceIndex].re * deltaCRe - coeff_B[referenceIndex].im * deltaCIm);
        double next_eps_im = (coeff_A[referenceIndex].re * deltaIm + coeff_A[referenceIndex].im * deltaRe) + 
                             (coeff_B[referenceIndex].re * deltaCIm + coeff_B[referenceIndex].im * deltaCRe);              
        deltaRe = next_eps_re; deltaIm = next_eps_im;
        referenceIndex += 100; iter += 100;
        zRe = reference[referenceIndex].re + deltaRe;
        zIm = reference[referenceIndex].im + deltaIm;
        if (zRe * zRe + zIm * zIm &gt;= ESCAPE_RADIUS_SQUARED) {
            deltaRe = backup_deltaRe; deltaIm = backup_deltaIm;
            referenceIndex = backup_refIdx; iter = backup_iter;
        } else {
            continue; 
        }
    }
    const double a = 2.0 * reference[referenceIndex].re + deltaRe;
    const double b = 2.0 * reference[referenceIndex].im + deltaIm;
    const double nextDeltaRe = a * deltaRe - b * deltaIm + deltaCRe;
    deltaIm = a * deltaIm + b * deltaRe + deltaCIm;
    deltaRe = nextDeltaRe;
    ++referenceIndex; ++iter;
    zRe = reference[referenceIndex].re + deltaRe;
    zIm = reference[referenceIndex].im + deltaIm;
    if (zRe * zRe + zIm * zIm &lt; deltaRe * deltaRe + deltaIm * deltaIm || referenceIndex == referenceLength - 1) {
        deltaRe = zRe; deltaIm = zIm; referenceIndex = 0;
    }
}
          const int remaining = MAX_ITER - iter;
          const uint8_t colorIndex = (remaining == 0) ? 255 : static_cast&lt;uint8_t&gt;(remaining % 254);
          const int paletteIndex = (colorIndex == 255) ? 255 : (colorIndex - PALETTE_FRAME + 255) % 255;
          blueSum += palette[paletteIndex][0];
          greenSum += palette[paletteIndex][1];
          redSum += palette[paletteIndex][2];
        }
      }
      const int samples = AA * AA;
      row[x * 3 + 0] = static_cast&lt;uint8_t&gt;(blueSum / samples);
      row[x * 3 + 1] = static_cast&lt;uint8_t&gt;(greenSum / samples);
      row[x * 3 + 2] = static_cast&lt;uint8_t&gt;(redSum / samples);
    }
    const int done = ++completedRows;
    if (done % 50 == 0 || done == HEIGHT) {
      std::fprintf(stderr, &quot;\rProgress: %d/%d rows (%.1f%%)&quot;, done, HEIGHT, 100.0 * done / HEIGHT);
    }
  }
  BMPHeader header;
  header.width = WIDTH; header.height = HEIGHT;
  header.sizeImage = static_cast&lt;uint32_t&gt;(image.size());
  header.size = header.sizeImage + 54;
  FILE * file = std::fopen(OUTPUT_FILE, &quot;wb&quot;);
  if (!file) { std::perror(OUTPUT_FILE); return 1; }
  std::fwrite(&amp;header, sizeof header, 1, file);
  std::fwrite(image.data(), 1, image.size(), file);
  std::fclose(file);
  std::fprintf(stderr, &quot;\nDone: \&quot;%s\&quot; saved in %.1f s\n&quot;, OUTPUT_FILE, omp_get_wtime() - startTime);
  return 0;
}
&lt;/syntaxhighlight&gt;

== Rendered Examples ==
&lt;gallery mode=&quot;packed&quot; heights=&quot;200&quot;&gt;
File:Mandelbrot Set Image 107.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 108.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 109.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 110.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
&lt;/gallery&gt;

== External Links ==
* [https://github.com/Divetoxx/Mandelbrot Official Mandelbrot CLI Repository on GitHub] — source code, documentation, and pre-compiled releases.
* [https://rosettacode.org/wiki/Mandelbrot_set#Perturbation_Theory Rosetta Code: Mandelbrot set Implementation] — C++ perturbation theory optimization showcased in the global code repository.


[[Category:Computer graphics]]
[[Category:Fractals]]</text>
      <sha1>0du3f4b59bh18h1q6079bwmcb7nihbe</sha1>
    </revision>
    <revision>
      <id>2820599</id>
      <parentid>2820598</parentid>
      <timestamp>2026-08-04T18:11:10Z</timestamp>
      <contributor>
        <username>Aokoroko</username>
        <id>2811879</id>
      </contributor>
      <comment>/* Key Features */</comment>
      <origin>2820599</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10816" sha1="eiivbmpzodoawe4k1u4bxc0dttocjj6" xml:space="preserve">== Introduction ==
This page contains the original C++ source code used to render high-precision fragments of the Mandelbrot set using perturbation theory and 8x8 Super-Sampling Anti-Aliasing (SSAA). Created by [[User:Aokoroko]].

== Key Features ==
* '''High-Precision Reference:''' The 1000-bit reference trajectory is computed exactly once per zoom layer.
* '''Hardware-Native Performance:''' Blazing-fast math for billions of pixels utilizing hardware-native double registers.
* '''Bilinear approximation:''' The calculation can be performed significantly faster by using bilinear approximation.
* When using double-precision floating-point numbers (on the order of 10⁻¹⁵), perturbation theory only allows you to zoom down to the '''10⁻³⁰⁸ level—no further.'''
* '''Innovative Algorithm:''' Revolutionary *Reference Reset to Zero* implementation.
* '''True 8x8 SSAA:''' Pristine, anti-aliased image quality with 64 independent samples per pixel.
* '''OpenMP Multi-threading:''' High-speed parallel computing to maximize CPU utilization.

== C++ Source Code ==

&lt;syntaxhighlight lang=&quot;cpp&quot;&gt;
#include &lt;atomic&gt;
#include &lt;cmath&gt;
#include &lt;cstdint&gt;
#include &lt;cstdio&gt;
#include &lt;vector&gt;
#include &lt;algorithm&gt;
#include &lt;mpfr.h&gt;
#include &lt;omp.h&gt;
using std::vector;
using std::min;
const char * CENTER_RE = &quot;-1.99999543561201124623198345433951143502785679245726844745821388800402678499411681518036306219179273434395557574279985918047221291197081186140687781560831995&quot;;
const char * CENTER_IM = &quot;-0.00000000000000000000000026198152173811047783694060060607013913873144250985383083459221663448338433592617272786772587281530484110756597337683912309313885172&quot;;
const char * VIEW_SIZE = &quot;1.15e-119&quot;;
const int WIDTH = 2160;
const int HEIGHT = 2160;
const int AA = 8;
const int MAX_ITER = 50000;
const double ESCAPE_RADIUS_SQUARED = 50000.0;
const int PALETTE_FRAME = 200;
const char * OUTPUT_FILE = &quot;Mandelbrot Set Image 112.bmp&quot;;
const mpfr_prec_t PRECISION_BITS = 1000;
struct Complex {
  double re;
  double im;
};
#pragma pack(push, 1)
struct BMPHeader {
  uint16_t type{0x4D42};
  uint32_t size{0};
  uint32_t reserved{0};
  uint32_t offBits{54};
  uint32_t structSize{40};
  int32_t width{0};
  int32_t height{0};
  uint16_t planes{1};
  uint16_t bitCount{24};
  uint32_t compression{0};
  uint32_t sizeImage{0};
  int32_t xPixelsPerMeter{2834};
  int32_t yPixelsPerMeter{2834};
  uint32_t colorsUsed{0};
  uint32_t colorsImportant{0};
};
#pragma pack(pop)
int main() {
  const double startTime = omp_get_wtime();
  const long rawWidth = static_cast&lt;long&gt;(WIDTH) * AA;
  const long rawHeight = static_cast&lt;long&gt;(HEIGHT) * AA;
  mpfr_t centerRe, centerIm, zReMp, zImMp, tmp1, tmp2, viewSizeMp;
  mpfr_inits2(PRECISION_BITS, centerRe, centerIm, zReMp, zImMp, tmp1, tmp2, viewSizeMp, static_cast&lt;mpfr_ptr&gt;(nullptr));
  mpfr_set_str(centerRe, CENTER_RE, 10, MPFR_RNDN);
  mpfr_set_str(centerIm, CENTER_IM, 10, MPFR_RNDN);
  mpfr_set_str(viewSizeMp, VIEW_SIZE, 10, MPFR_RNDN);
  const double sampleStep = mpfr_get_d(viewSizeMp, MPFR_RNDN) / rawWidth;
  vector&lt;Complex&gt; referenceOrbit;
  referenceOrbit.reserve(MAX_ITER + 200);
  mpfr_set_ui(zReMp, 0, MPFR_RNDN);
  mpfr_set_ui(zImMp, 0, MPFR_RNDN);
  for (int iter = 0; iter &lt;= MAX_ITER + 150; ++iter) {
    Complex z{mpfr_get_d(zReMp, MPFR_RNDN), mpfr_get_d(zImMp, MPFR_RNDN)};
    referenceOrbit.push_back(z);
    if (z.re * z.re + z.im * z.im &gt; ESCAPE_RADIUS_SQUARED &amp;&amp; iter &gt; MAX_ITER) {
      break;
    }
    mpfr_mul(tmp1, zReMp, zImMp, MPFR_RNDN);
    mpfr_sqr(tmp2, zReMp, MPFR_RNDN);
    mpfr_sqr(zReMp, zImMp, MPFR_RNDN);
    mpfr_sub(zReMp, tmp2, zReMp, MPFR_RNDN);
    mpfr_add(zReMp, zReMp, centerRe, MPFR_RNDN);
    mpfr_mul_2ui(tmp1, tmp1, 1, MPFR_RNDN);
    mpfr_add(zImMp, tmp1, centerIm, MPFR_RNDN);
  }
  const int referenceLength = static_cast&lt;int&gt;(referenceOrbit.size());
  mpfr_clears(centerRe, centerIm, zReMp, zImMp, tmp1, tmp2, viewSizeMp, static_cast&lt;mpfr_ptr&gt;(nullptr));
  std::fprintf(stderr, &quot;Reference orbit: %d points\n&quot;, referenceLength);
  std::fprintf(stderr, &quot;Precomputing skip100 matrices...\n&quot;);
  vector&lt;Complex&gt; coeff_A(referenceLength, {1.0, 0.0});
  vector&lt;Complex&gt; coeff_B(referenceLength, {0.0, 0.0});
  vector&lt;double&gt; rad_R(referenceLength, 2.0);
  #pragma omp parallel for
  for (int i = 0; i &lt; MAX_ITER; ++i) {
    double min_r = 2.0;
    for (int k = 0; k &lt; 100; ++k) {
      if (i + k &gt;= referenceLength) break;
      double r_re = referenceOrbit[i + k].re;
      double r_im = referenceOrbit[i + k].im;
      double abs_s = std::sqrt(r_re * r_re + r_im * r_im);
      double aS = (abs_s &lt; 2.0) ? abs_s : 0.0;
      min_r = min(min_r, aS);
      double next_A_re = 2.0 * (r_re * coeff_A[i].re - r_im * coeff_A[i].im);
      double next_A_im = 2.0 * (r_re * coeff_A[i].im + r_im * coeff_A[i].re);
      double next_B_re = 2.0 * (r_re * coeff_B[i].re - r_im * coeff_B[i].im) + 1.0;
      double next_B_im = 2.0 * (r_re * coeff_B[i].im + r_im * coeff_B[i].re);
      coeff_A[i].re = next_A_re; coeff_A[i].im = next_A_im;
      coeff_B[i].re = next_B_re; coeff_B[i].im = next_B_im;
    }
    rad_R[i] = min_r;
  }
  const double PI = 3.14159265358979323846;
  uint8_t palette[256][3];
  for (int i = 0; i &lt; 255; ++i) {
    palette[i][0] = static_cast&lt;uint8_t&gt;(std::lround(127.0 + 127.0 * std::cos(2.0 * PI * i / 255.0)));
    palette[i][1] = static_cast&lt;uint8_t&gt;(std::lround(127.0 + 127.0 * std::sin(2.0 * PI * i / 255.0)));
    palette[i][2] = palette[i][1];
  }
  palette[255][0] = 255; palette[255][1] = 255; palette[255][2] = 255;
  const int rowBytes = (WIDTH * 3 + 3) &amp; ~3;
  vector&lt;uint8_t&gt; image(static_cast&lt;size_t&gt;(rowBytes) * HEIGHT, 0);
  std::atomic&lt;int&gt; completedRows{0};
  const Complex * reference = referenceOrbit.data();
  #pragma omp parallel for schedule(dynamic)
  for (int y = 0; y &lt; HEIGHT; ++y) {
    uint8_t * row = image.data() + static_cast&lt;size_t&gt;(y) * rowBytes;
    for (int x = 0; x &lt; WIDTH; ++x) {
      unsigned blueSum = 0; unsigned greenSum = 0; unsigned redSum = 0;
      for (int sampleY = 0; sampleY &lt; AA; ++sampleY) {
        const double deltaCIm = (static_cast&lt;long&gt;(y) * AA + sampleY - rawHeight / 2) * sampleStep;
        for (int sampleX = 0; sampleX &lt; AA; ++sampleX) {
          const double deltaCRe = (static_cast&lt;long&gt;(x) * AA + sampleX - rawWidth / 2) * sampleStep;
          double deltaRe = 0.0; double deltaIm = 0.0;
          double zRe = 0.0;     double zIm = 0.0;
          int referenceIndex = 0;
          int iter = 0;
while (iter &lt; MAX_ITER) {
    if (zRe * zRe + zIm * zIm &gt;= ESCAPE_RADIUS_SQUARED) {
        break;
    }
    double eps_abs2 = deltaRe * deltaRe + deltaIm * deltaIm;
    double limit_r2 = 1e-60 * rad_R[referenceIndex] * rad_R[referenceIndex];
    if (eps_abs2 &lt; limit_r2 &amp;&amp; (referenceIndex + 100 &lt; referenceLength - 1) &amp;&amp; (iter + 100 &lt; MAX_ITER)) {
        double backup_deltaRe = deltaRe; double backup_deltaIm = deltaIm;
        int backup_refIdx = referenceIndex; int backup_iter = iter;
        double next_eps_re = (coeff_A[referenceIndex].re * deltaRe - coeff_A[referenceIndex].im * deltaIm) + 
                             (coeff_B[referenceIndex].re * deltaCRe - coeff_B[referenceIndex].im * deltaCIm);
        double next_eps_im = (coeff_A[referenceIndex].re * deltaIm + coeff_A[referenceIndex].im * deltaRe) + 
                             (coeff_B[referenceIndex].re * deltaCIm + coeff_B[referenceIndex].im * deltaCRe);              
        deltaRe = next_eps_re; deltaIm = next_eps_im;
        referenceIndex += 100; iter += 100;
        zRe = reference[referenceIndex].re + deltaRe;
        zIm = reference[referenceIndex].im + deltaIm;
        if (zRe * zRe + zIm * zIm &gt;= ESCAPE_RADIUS_SQUARED) {
            deltaRe = backup_deltaRe; deltaIm = backup_deltaIm;
            referenceIndex = backup_refIdx; iter = backup_iter;
        } else {
            continue; 
        }
    }
    const double a = 2.0 * reference[referenceIndex].re + deltaRe;
    const double b = 2.0 * reference[referenceIndex].im + deltaIm;
    const double nextDeltaRe = a * deltaRe - b * deltaIm + deltaCRe;
    deltaIm = a * deltaIm + b * deltaRe + deltaCIm;
    deltaRe = nextDeltaRe;
    ++referenceIndex; ++iter;
    zRe = reference[referenceIndex].re + deltaRe;
    zIm = reference[referenceIndex].im + deltaIm;
    if (zRe * zRe + zIm * zIm &lt; deltaRe * deltaRe + deltaIm * deltaIm || referenceIndex == referenceLength - 1) {
        deltaRe = zRe; deltaIm = zIm; referenceIndex = 0;
    }
}
          const int remaining = MAX_ITER - iter;
          const uint8_t colorIndex = (remaining == 0) ? 255 : static_cast&lt;uint8_t&gt;(remaining % 254);
          const int paletteIndex = (colorIndex == 255) ? 255 : (colorIndex - PALETTE_FRAME + 255) % 255;
          blueSum += palette[paletteIndex][0];
          greenSum += palette[paletteIndex][1];
          redSum += palette[paletteIndex][2];
        }
      }
      const int samples = AA * AA;
      row[x * 3 + 0] = static_cast&lt;uint8_t&gt;(blueSum / samples);
      row[x * 3 + 1] = static_cast&lt;uint8_t&gt;(greenSum / samples);
      row[x * 3 + 2] = static_cast&lt;uint8_t&gt;(redSum / samples);
    }
    const int done = ++completedRows;
    if (done % 50 == 0 || done == HEIGHT) {
      std::fprintf(stderr, &quot;\rProgress: %d/%d rows (%.1f%%)&quot;, done, HEIGHT, 100.0 * done / HEIGHT);
    }
  }
  BMPHeader header;
  header.width = WIDTH; header.height = HEIGHT;
  header.sizeImage = static_cast&lt;uint32_t&gt;(image.size());
  header.size = header.sizeImage + 54;
  FILE * file = std::fopen(OUTPUT_FILE, &quot;wb&quot;);
  if (!file) { std::perror(OUTPUT_FILE); return 1; }
  std::fwrite(&amp;header, sizeof header, 1, file);
  std::fwrite(image.data(), 1, image.size(), file);
  std::fclose(file);
  std::fprintf(stderr, &quot;\nDone: \&quot;%s\&quot; saved in %.1f s\n&quot;, OUTPUT_FILE, omp_get_wtime() - startTime);
  return 0;
}
&lt;/syntaxhighlight&gt;

== Rendered Examples ==
&lt;gallery mode=&quot;packed&quot; heights=&quot;200&quot;&gt;
File:Mandelbrot Set Image 107.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 108.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 109.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 110.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
&lt;/gallery&gt;

== External Links ==
* [https://github.com/Divetoxx/Mandelbrot Official Mandelbrot CLI Repository on GitHub] — source code, documentation, and pre-compiled releases.
* [https://rosettacode.org/wiki/Mandelbrot_set#Perturbation_Theory Rosetta Code: Mandelbrot set Implementation] — C++ perturbation theory optimization showcased in the global code repository.


[[Category:Computer graphics]]
[[Category:Fractals]]</text>
      <sha1>eiivbmpzodoawe4k1u4bxc0dttocjj6</sha1>
    </revision>
  </page>
  <page>
    <title>The John Snow Prediabetes Institute</title>
    <ns>0</ns>
    <id>330494</id>
    <revision>
      <id>2820565</id>
      <parentid>2819642</parentid>
      <timestamp>2026-08-04T13:18:50Z</timestamp>
      <contributor>
        <username>NDM2024</username>
        <id>2984088</id>
      </contributor>
      <minor />
      <origin>2820565</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5091" sha1="463gy66oo9sxmamigf3krqdyfu7rdch" xml:space="preserve">
'''&lt;big&gt;The John Snow Prediabetes Institute.&lt;/big&gt;'''

https://w.wiki/Skm7

The John Snow prediabetes Institute is an international research network focused on prediabetes (prevention) remission, early risk identification, and metabolic health education.

&lt;big&gt;Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer and self-evaluation of diet and physical activity. Educational materials is available from the international diabetes organisations e.g from the ADA: &lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt;[[File:Lifestyle Medicine Pillars.png|300px|right|&lt;big&gt;The focus of Lifestyle Medicine is on these 6 pillars.&lt;/big&gt;]]
[[File:Cholera in London 1866.gif|thumb|250px|&lt;big&gt;Map of a later cholera outbreak in London, in 1866&lt;/big&gt;]] [[File:Choleramaplondon1866.png|thumb|right|250px|&lt;big&gt;Legend for the map above&lt;/big&gt;]]&lt;big&gt;1. '&lt;nowiki/&gt;'''Prevalence studies''''&lt;/big&gt;
&lt;big&gt;1.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.3 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.4. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.5. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;&lt;br /&gt;
2. '''Intervention studies''' Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;- General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;- John Snow Institute bylaws &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;3. '''Publications and pptx''' 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;4. '''Prediabetes-Remission Research Network:'''&lt;/big&gt;
&lt;small&gt;Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey (Coordinator); Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark&lt;/small&gt; 

==References==

&lt;references /&gt;
[[Category:Prediabetes ]]Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
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'''&lt;big&gt;The John Snow Prediabetes Institute.&lt;/big&gt;'''

https://w.wiki/Skm7

The John Snow prediabetes Institute is an international research network focused on prediabetes (prevention) remission, early risk identification, and metabolic health education.

&lt;big&gt;Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer and self-evaluation of diet and physical activity. Educational materials is available from the international diabetes organisations e.g from the ADA: &lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt;[[File:Lifestyle Medicine Pillars.png|300px|right|&lt;big&gt;The focus of Lifestyle Medicine is on these 6 pillars.&lt;/big&gt;]]
[[File:Cholera in London 1866.gif|thumb|250px|&lt;big&gt;Map of a later cholera outbreak in London, in 1866&lt;/big&gt;]] [[File:Choleramaplondon1866.png|thumb|right|250px|&lt;big&gt;Legend for the map above&lt;/big&gt;]]&lt;big&gt;1. '&lt;nowiki/&gt;'''Prevalence studies''''&lt;/big&gt;
&lt;big&gt;1.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;&lt;br /&gt;
2. '''Intervention studies''' Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;- General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;- John Snow Institute bylaws &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;3. '''Publications and pptx''' 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;4. '''Prediabetes-Remission Research Network:'''&lt;/big&gt;
&lt;small&gt;Cordinator and Director MBA Christian Acheampong, Turkey, Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark&lt;/small&gt; 

==References==

&lt;references /&gt;
[[Category:Prediabetes ]]Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
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{{Information
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== Licensing ==
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== Licensing ==
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&lt;div align=center&gt;Edit the title and sub-title to match the wording (and casing) in the [[Motivation and emotion/Book/2025|2026 list of topics]].&lt;br&gt;[[Motivation and emotion/About/Staff|Seek approval]] for any changes.&lt;br&gt;Do not include your name (authorship is as per [[Special:History/{{PAGENAME}}|the page history]]).&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Imagine this ... or Scenario ... or Case study or ... ?)

Start with an engaging [[#Scenarios|scenario, example, or case study]] which illustrates the problem and engages reader interest.

Present the scenario in a [[#Feature box|feature box]]. To change the box colour:
# Edit source
# Change &quot;theme=3&quot; to another number

Include an image and cite it (e.g., see Figure 1).

{{RoundBoxBottom}}

The Overview section should provide:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topc''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.

This template provides key headings, examples, and tips for each section. Gradually remove this generic information as the chapter develops. It is OK to retain some of the template material for the topic development, but it should all be removed for the final book chapter.

Key resources:
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] explains about how to edit
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]

{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask [[w:Open-ended question|open-ended]] focus questions. For example:
* Is there a relationship between weather and criminal behaviour? (closed-ended)
* What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Headings==
Use this heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings, but:
** avoid having only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* See also
* References
* External links

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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== Licensing ==
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&lt;div align=center&gt;Edit the title and sub-title to match the wording (and casing) in the [[Motivation and emotion/Book/2025|2026 list of topics]].&lt;br&gt;[[Motivation and emotion/About/Staff|Seek approval]] for any changes.&lt;br&gt;Do not include your name (authorship is as per [[Special:History/{{PAGENAME}}|the page history]]).&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Imagine this ... or Scenario ... or Case study or ... ?)

Start with an engaging [[#Scenarios|scenario, example, or case study]] which illustrates the problem and engages reader interest.

Present the scenario in a [[#Feature box|feature box]]. To change the box colour:
# Edit source
# Change &quot;theme=3&quot; to another number

Include an image and cite it (e.g., see Figure 1).

{{RoundBoxBottom}}

The Overview section should provide:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topc''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.

This template provides key headings, examples, and tips for each section. Gradually remove this generic information as the chapter develops. It is OK to retain some of the template material for the topic development, but it should all be removed for the final book chapter.

Key resources:
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] explains about how to edit
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]

{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask [[w:Open-ended question|open-ended]] focus questions. For example:
* Is there a relationship between weather and criminal behaviour? (closed-ended)
* What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Headings==
Use this heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings, but:
** avoid having only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* See also
* References
* External links

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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The following instructions should be used to guide the development of the book chapter.

=== Theme ===

* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

=== Audience ===

* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a science communication exercise.

=== Wikiversity ===

* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

=== Topic ===

* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

=== Collaboration and feedback ===

* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a UCLearn discussion thread)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

=== Length (word count) ===

* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this Word Counter (Google Chrome Extension) or paste the URL into Webpage Word Counter (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[Motivation and emotion/Assessment/Chapter/Word count|these suggestions]]

=== Submission ===

* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via UCLearn

== Marking criteria ==
Book chapters will be marked against the following criteria.

=== Overview (5%) ===

* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

=== Theory (20%) ===

* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

=== Research (25%) ===

* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* Critical analyse the key research findings, including limitations and implications

=== Integration (10%) ===

* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

=== Conclusion (5%) ===

* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

=== Style (20%) ===

* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[Motivation and emotion/Assessment/Chapter/Readability|Readable]] for a layperson interested in psychological science
** Address the [[Motivation and emotion/Assessment/Chapter#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[Motivation and emotion/Assessment/Chapter/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use sentence casing throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** Narrative point of view[1]: In the main text, use 3rd person perspective (e.g., &quot;it&quot;, &quot;they&quot;). Where asides are used, such as examples, case studies, and feature boxes, 1st person perspective (e.g., &quot;I&quot; and &quot;we&quot;) and/or 2nd person perspective (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or wiki citation style. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or wiki citation style. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

=== Learning features (5%) ===

* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** emotion involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &lt;nowiki&gt;&amp;lt;nowiki&amp;gt;&lt;/nowiki&gt;[[Motivation and emotion/Assessment/Chapter#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter

=== Social contribution (10%) ===

* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[Motivation and emotion/Assessment/Chapter/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk pages]]
** '''Media uploads''': Upload free-to-use educational images to Wikimedia Commons
** '''UCLearn discussion posts'''
* '''Evidence''':
** Provide a numbered list with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.
** More info: [[Motivation and emotion/Assessment/Chapter/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00</text>
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The following instructions should be used to guide the development of the book chapter.

=== Theme ===

* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

=== Audience ===

* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a science communication exercise.

=== Wikiversity ===

* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

=== Topic ===

* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

=== Collaboration and feedback ===

* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a UCLearn discussion thread)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

=== Length (word count) ===

* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this Word Counter (Google Chrome Extension) or paste the URL into Webpage Word Counter (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[Motivation and emotion/Assessment/Chapter/Word count|these suggestions]]

=== Submission ===

* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via UCLearn

== Marking criteria ==
Book chapters will be marked against the following criteria.

=== Overview (5%) ===

* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

=== Theory (20%) ===

* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

=== Research (25%) ===

* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* Critical analyse the key research findings, including limitations and implications

=== Integration (10%) ===

* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

=== Conclusion (5%) ===

* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

=== Style (20%) ===

* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[Motivation and emotion/Assessment/Chapter/Readability|Readable]] for a layperson interested in psychological science
** Address the [[Motivation and emotion/Assessment/Chapter#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[Motivation and emotion/Assessment/Chapter/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use sentence casing throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** Narrative point of view[1]: In the main text, use 3rd person perspective (e.g., &quot;it&quot;, &quot;they&quot;). Where asides are used, such as examples, case studies, and feature boxes, 1st person perspective (e.g., &quot;I&quot; and &quot;we&quot;) and/or 2nd person perspective (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or wiki citation style. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or wiki citation style. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

=== Learning features (5%) ===

* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** emotion involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &lt;nowiki&gt;&amp;lt;nowiki&amp;gt;&lt;/nowiki&gt;[[Motivation and emotion/Assessment/Chapter#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter

=== Social contribution (10%) ===

* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[Motivation and emotion/Assessment/Chapter/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk pages]]
** '''Media uploads''': Upload free-to-use educational images to Wikimedia Commons
** '''UCLearn discussion posts'''
* '''Evidence''':
** Provide a numbered list with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.
** More info: [[Motivation and emotion/Assessment/Chapter/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00</text>
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    </revision>
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      </contributor>
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The following instructions should be used to guide the development of the book chapter.

=== Theme ===

* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

=== Audience ===

* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a science communication exercise.

=== Wikiversity ===

* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

=== Topic ===

* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

=== Collaboration and feedback ===

* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a UCLearn discussion thread)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

=== Length (word count) ===

* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this Word Counter (Google Chrome Extension) or paste the URL into Webpage Word Counter (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[Motivation and emotion/Assessment/Chapter/Word count|these suggestions]]

=== Submission ===

* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via UCLearn

== Marking criteria ==
Book chapters will be marked against the following criteria.

=== Overview (5%) ===

* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

=== Theory (20%) ===

* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

=== Research (25%) ===

* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* Critical analyse the key research findings, including limitations and implications

=== Integration (10%) ===

* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

=== Conclusion (5%) ===

* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

=== Style (20%) ===

* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[Motivation and emotion/Assessment/Chapter/Readability|Readable]] for a layperson interested in psychological science
** Address the [[Motivation and emotion/Assessment/Chapter#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[Motivation and emotion/Assessment/Chapter/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use sentence casing throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** Narrative point of view[1]: In the main text, use 3rd person perspective (e.g., &quot;it&quot;, &quot;they&quot;). Where asides are used, such as examples, case studies, and feature boxes, 1st person perspective (e.g., &quot;I&quot; and &quot;we&quot;) and/or 2nd person perspective (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or wiki citation style. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or wiki citation style. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

=== Learning features (5%) ===

* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** emotion involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &lt;nowiki&gt;&amp;lt;nowiki&amp;gt;&lt;/nowiki&gt;[[Motivation and emotion/Assessment/Chapter#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter

=== Social contribution (10%) ===

* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[Motivation and emotion/Assessment/Chapter/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk pages]]
** '''Media uploads''': Upload free-to-use educational images to Wikimedia Commons
** '''UCLearn discussion posts'''
* '''Evidence''':
** Provide a numbered list with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.
** More info: [[Motivation and emotion/Assessment/Chapter/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00</text>
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      <comment>Welcome</comment>
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      <text bytes="2325" sha1="p8fxmo758eg1kkwpeq3wyurxt7l7vvy" xml:space="preserve">==Welcome==
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See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:07, 5 August 2026 (UTC)&lt;/div&gt;
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      <comment>Welcome</comment>
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* [[Wikiversity:Browse|Browse]] or visit an educational level portal:&lt;br&gt;[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
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To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:07, 5 August 2026 (UTC)&lt;/div&gt;
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  <page>
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      <timestamp>2026-08-05T06:11:43Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
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      <comment>Welcome</comment>
      <origin>2820659</origin>
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      <text bytes="2325" sha1="04ge42zzlth1v4t9tyykkiwdoeaovbi" xml:space="preserve">==Welcome==
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To find your way around, check out:
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* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:&lt;br&gt;[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
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To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:11, 5 August 2026 (UTC)&lt;/div&gt;
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Welcome</comment>
      <origin>2820669</origin>
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      <text bytes="2325" sha1="p3e3kzm0hw6hf26yqhay4nv8nlst1bs" xml:space="preserve">==Welcome==
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You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]].

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To find your way around, check out:
&lt;!-- The Left column --&gt;
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* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:&lt;br&gt;[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
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To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:27, 5 August 2026 (UTC)&lt;/div&gt;
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